Cauchy Moments Research Articles

Theoretical determination of polarizability and magnetic susceptibility of neon

We report theoretical determination of the dipole polarizability of the neon atom, including its frequency dependence. Corrections for the relativistic, quantum electrodynamics, finite nuclear mass, and finite nuclear size effects are taken into account. We obtain the value $\alpha_0=2.66080(36)$ for the static polarizability, and $\alpha_2=2.850(7)$ and $\alpha_4=4.932(14)$ for the first two polarizability dispersion coefficients (Cauchy moments); all values are in atomic units (a.u.). In the case of static polarizability, our result agrees with the best experimental determination [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)], but our estimated uncertainty is significantly larger. For the dispersion coefficients, the results obtained in this work appear to be the most accurate to date overall compared to published theoretical and experimental data. We also calculated the static magnetic susceptibility of the neon atom, needed to obtain the refractive index of gaseous neon. Our result, $\chi_0 = -8.484(19) \cdot 10^{-5}$ a.u., is about 9% larger in absolute value than the recommended experimental value [CRC Handbook of Chemistry and Physics, CRC Press, 2019, p. 4-145].

Ab initio calculations of static dipole polarizabilities and Cauchy moments for the halomethanes, CHmClnF4−m−n

Coupled-cluster calculations of the static electronic dipole polarizabilities and Cauchy moments are reported for all 15 halomethanes CHmClnF4−m−n. Comparison with available experimental static polarizabilities is made. Excluding three experimental values which seem to be in error, the mean absolute deviation between the CCSD(T) values and experiment is a rather satisfactory 0.5 atomic units for the remaining 11 halomethanes. More experimental work is needed for the polarizabilities of CHCl2F, CCl3F, and CH2Cl2. Additivity approximations work moderately well for α=S(−2) and progressively less well for S(−2k−2) as k increases.

On the dipole polarisability and dipole sum rules of ozone

Ab initio calculations of the dipole polarisability and other Cauchy moments of the dipole oscillator strength distribution (DOSD) of ozone are reported to help resolve discrepancies between theory and experiment. A number of coupled-cluster methods based on a Hartree–Fock reference function, multiconfiguration-reference configuration interaction methods, and perturbatively corrected, complete-active-space self-consistent field methods are used. The C DOSD of Kumar and Thakkar is probably preferable to their B1 distribution. Our best estimate of the mean polarisability is atomic units.

Dispersion coefficients and frequency-dependent polarizabilities

Extensions of previous methods for obtaining two- and three-body dispersion coefficients are presented. Many-dimensional arrays in inner projections are utilized. The required matrix elements are expressed in terms of atomic frequency (Cauchy) moments. Approximate dynamic polarizabilities are derived using generalized inner projections, and a number of theorems for definite operators are extended to more general cases. Numerical results for both two- and three-body dispersion coefficients are reported.

Dynamic polarizability, Cauchy moments, and the optical absorption spectrum of liquid water: A sequential molecular dynamics/quantum mechanical approach

The dynamic polarizability and optical absorption spectrum of liquid water in the 6-15 eV energy range are investigated by a sequential molecular dynamics (MD)/quantum mechanical approach. The MD simulations are based on a polarizable model for liquid water. Calculation of electronic properties relies on time-dependent density functional and equation-of-motion coupled-cluster theories. Results for the dynamic polarizability, Cauchy moments, S(-2), S(-4), S(-6), and dielectric properties of liquid water are reported. The theoretical predictions for the optical absorption spectrum of liquid water are in good agreement with experimental information.

Linear response at the 4-component relativistic density-functional level: application to the frequency-dependent dipole polarizability of Hg, AuH and PtH2

Abstract We report the implementation and application of linear response density-functional theory (DFT) based on the 4-component relativistic Dirac–Coulomb Hamiltonian. The theory is cast in the language of second quantization and is based on the quasienergy formalism (Floquet theory), replacing the initial state dependence of the Runge–Gross theorem by periodic boundary conditions. Contradictions in causality and symmetry of the time arguments are thereby avoided and the exchange-correlation potential and kernel can be expressed as functional derivatives of the quasienergy. We critically review the derivation of the quasienergy analogues of the Hohenberg–Kohn theorem and the Kohn–Sham formalism and discuss the nature of the quasienergy exchange-correlation functional. Structure is imposed on the response equations in terms of Hermiticity and time-reversal symmetry. It is observed that functionals of spin and current densities, corresponding to time-antisymmetric operators, contribute to frequency-dependent and not static electric properties. Physically, this follows from the fact that only a time-dependent electric field creates a magnetic field. It is furthermore observed that hybrid functionals enhance spin polarization since only exact exchange contributes to anti-Hermitian trial vectors. We apply 4-component relativistic linear response DFT to the calculation of the frequency-dependent polarizability of the isoelectronic series Hg, AuH and PtH2. Unlike for the molecules, the effect of electron correlation on the polarizability of the mercury atom is very large, about 25%. We observe a remarkable performance of the local-density approximation (LDA) functional in reproducing the experimental frequency-dependent polarizability of this atom, clearly superior to that of the BLYP and B3LYP functionals. This allows us to extract Cauchy moments (S(−4) = 382.82 and S(−6) = 6090.89 a.u.) that we believe are superior to experiment since we go to higher order in the Cauchy moment expansion.

Calculation of optical characteristics of atoms with a closed shell by the Hartree-Fock-Roothaan method

The transition frequencies, oscillator strengths, and Cauchy moments of the dipole dynamic polarizability are calculated for the Ar, Ca, Sr, and Xe atoms and their isoelectronic sequences.

Polarization propagator calculations of the polarizability tensor at imaginary frequencies and long-range interactions for the noble gases and n-alkanes

The linear polarization propagator has been computed at imaginary frequencies for He, Ne, Ar, and Kr as well as for the n-alkanes including heptane and its smaller members. It is shown that an effective and direct evaluation of the polarization propagator using standard electronic structure first principle methods can be achieved on the whole imaginary axis without expanding the polarizability in a series of the Cauchy moments. The linear response equation will be complex in this case, but an effective algorithm can be constructed so that the computational cost parallels that of the real propagator. Calculations of the polarizability tensor are used to determine the Casimir–Polder interaction potentials for the molecules under consideration. Theoretical results for the C6 dispersion coefficient are compared with accurate experimental data, and it is shown that results for the extended n-alkanes obtained with density functional theory and the hybrid B3LYP exchange correlation functional are in excellent agreement with experiment. At the same level of theory, on the other hand, there are significant discrepancies for the noble gas atoms. The electron correlation contribution to C6 is less than 9% for the n-alkanes and decreases with the size of the system.

On the required shape corrections to the local density and generalized gradient approximations to the Kohn–Sham potentials for molecular response calculations of (hyper)polarizabilities and excitation energies

It is well known that shape corrections have to be applied to the local-density (LDA) and generalized gradient (GGA) approximations to the Kohn–Sham exchange–correlation potential in order to obtain reliable response properties in time dependent density functional theory calculations. Here we demonstrate that it is an oversimplified view that these shape corrections concern primarily the asymptotic part of the potential, and that they affect only Rydberg type transitions. The performance is assessed of two shape-corrected Kohn–Sham potentials, the gradient-regulated asymptotic connection procedure applied to the Becke–Perdew potential (BP–GRAC) and the statistical averaging of (model) orbital potentials (SAOP), versus LDA and GGA potentials, in molecular response calculations of the static average polarizability α, the Cauchy coefficient S−4, and the static average hyperpolarizability β. The nature of the distortions of the LDA/GGA potentials is highlighted and it is shown that they introduce many spurious excited states at too low energy which may mix with valence excited states, resulting in wrong excited state compositions. They also lead to wrong oscillator strengths and thus to a wrong spectral structure of properties like the polarizability. LDA, Becke–Lee–Yang–Parr (BLYP), and Becke–Perdew (BP) characteristically underestimate contributions to α and S−4 from bound Rydberg-type states and overestimate those from the continuum. Cancellation of the errors in these contributions occasionally produces fortuitously good results. The distortions of the LDA, BLYP, and BP spectra are related to the deficiencies of the LDA/GGA potentials in both the bulk and outer molecular regions. In contrast, both SAOP and BP-GRAC potentials produce high quality polarizabilities for 21 molecules and also reliable Cauchy moments and hyperpolarizabilities for the selected molecules. The analysis for the N2 molecule shows, that both SAOP and BP–GRAC yield reliable energies ωi and oscillator strengths fi of individual excitations, so that they reproduce well the spectral structure of α and S−4.

Comment on “Polarizabilities of the rare-gas homonuclear diatoms” [J. Chem. Phys. 111, 6316 (1999)

The collision-induced polarizability anisotropy γ(R) of the rare-gas homonuclear diatoms is represented in terms of the Cauchy moments S(−2k) of the single atoms. Except for neon, very good agreement is found between this new representation and other analytic forms given in the literature.

The effect of intermolecular interactions on the electric properties of helium and argon. I.Ab initiocalculation of the interaction induced polarizability and hyperpolarizability in He2 and Ar2

The frequency-dependent interaction induced polarizabilities and second hyperpolarizabilities are calculated for He2 at the coupled cluster singles and doubles and full configuration interaction levels and for Ar2 at the coupled cluster singles and doubles level. The frequency-dependence is approximated by a power series to second-order in the frequency arguments using Cauchy moments and hyperpolarizability dispersion coefficients. Using large correlation consistent basis sets, results close to the basis set limit are obtained. The computed curves for the interaction induced (hyper-) polarizabilities are tabulated for a range of internuclear distances. The data are employed in a companion paper to make for the first time a direct comparison between the experimentally determined pressure dependence of an ESHG hyperpolarizability and ab initio calculated hyperpolarizability second virial coefficients.

The electric-field-gradient-induced birefringence of Helium, Neon, Argon, and SF6

The results of an investigation of the electric-field-gradient-induced birefringence of the noble gases He, Ne, Ar and of the octahedral system SF6 are presented. For all these systems the induced anisotropy of the refractive index is related to the sole temperature-independent contribution b(ω), a combination of quadratic response functions. These hyperpolarizabilities, as well as their dispersion coefficients to seventh order, are calculated at the optical experimental frequency by the CCSD response approach. The dispersion coefficients are used to determine the curve of b(ω) in the normal dispersion region. We show that for atoms the dipole–dipole–quadrupole and the dipole–dipole–magnetic dipole contributions to the dispersion coefficients of b(ω) have intimate relations to the Cauchy moments and to the dispersion coefficients for the Verdet and the Cotton–Mouton constants.

Ab initio calculation of the frequency-dependent interaction induced hyperpolarizability of Ar2

The frequency-dependent interaction induced polarizability and second hyperpolarizability of the argon dimer are computed for a range of internuclear distances employing the coupled cluster singles and doubles response approach. The frequency dependence of the interaction-induced properties is treated through second order in the frequency arguments using expansions in Cauchy moments and hyperpolarizability dispersion coefficients. The dielectric, the refractivity, the Kerr and the hyperpolarizability second virial coefficients are computed for a range of temperatures employing a recent accurate ab initio potential for the ground state of the argon dimer. For most of the computed virial coefficients good agreement is obtained between the present ab initio results and the available experimental data.

Dispersion formulas for the second hyperpolarizability components γ||, γ⊥ and γK

The frequency-dependence of the isotropic parallel average γ || of the second hyperpolarizability can be expanded in a compact way using dispersion coefficients, which are independent of the optical process, if three effective frequencies are introduced. We propose similar expansions for the second hyperpolarizability components γ ⊥ and γ K and give explicit expressions for the hyperpolarizability dispersion coefficients in terms of the derivatives of the hyperpolarizabilities with respect to their frequency arguments. The relation of the hyperpolarizability dispersion coefficients to the Cauchy moments of linear response theory is discussed and comparison is made with previously proposed dispersion formulas.

Correlated, Static and Dynamic Polarizabilities of Small Molecules. Comparison of Four “Black Box” Methods

Molecular static and dynamic polarizabilities for thirteen small molecules have been calculated using four “black box” ab initio methods, the random phase approximation, RPA, the second-order polarization propagator approximation, SOPPA, the second-order polarization propagator approximation with coupled cluster singles and doubles amplitudes, SOPPA(CCSD), and the coupled cluster singles and doubles linear response function method, CCSDLR. The frequency dependence of the polarizabilities is given in terms of the dipole oscillator strength sum rules or Cauchy moments S(-4) and S(-6). Two basis sets were employed, Sadlej’s medium size polarized basis set and Dunning’s correlation consistent basis set of triple- œ quality augmented by two diffuse functions of each angular momentum (daug-cc-pVTZ). The results are compared to other theoretical results as well as to experimental values for the static polarizabilities, polarizability anisotropies, and Cauchy moments. Frequency-dependent polarizabilities and polarizability anisotropies, calculated at the CCSDLR level using the daug-cc-pVTZ basis set, are presented for five typical laser frequencies. The molecular dipole polarizability enters into the description of many physical and chemical processes, such as the scattering of light by molecules, and intermolecular interactions. Calculated polarizabilities are often used in the verification of experimental data and in the prediction of properties of new chemical species. An accuracy of a few percent in the calculated values is necessary for this purpose. Over the years several methods for the calculation of molecular properties have emerged. Among these are correlated methods, i.e. methods trying to improve on the Hartree-Fock approximation by perturbation theory or a multiconfigurational ansatz, as well as density functional theory (DFT) methods. Most of these methods, however, are only capable of calculating static properties like the static molecular polarizability, excitation energies, and transition moments. A direct comparison of calculated and experimental polarizabilities requires the ability to calculate frequency-dependent polarizabilities since experiments are mostly performed at nonzero frequencies. In an earlier study 1 the performance of some perturbation theory methods in the calculation of static polarizabilities was investigated. In this work, calculations of static and dynamic molecular polarizabilities are presented using four different “black box” methods, that is, methods where the only choices to be made are of the basis set and molecular geometry. These methods are in contrast to multiconfigurational methods where the selection of configurations to be included in the wave function requires considerable experience and might even become impossible for larger molecules. The black box methods, on the other hand, are relatively easy to use also by nonexperts, and their application is, apart from hardware limitations, not restricted to small molecules.

Cauchy moments and dispersion coefficients using coupled cluster linear response theory

Expressions for the even Cauchy moments for nonvariational methods have been derived using the time-averaged quasienergy Lagrangian technique. The expressions obtained require the solution of linear equations but do not involve a sum over individual excited-state contributions. An implementation is reported for the coupled cluster models CCS, CC2, and CCSD and calculations have been performed for the Cauchy moments and the Verdet and Cotton–Mouton constants of the Ne atom and for the C6 dispersion coefficient of the Ne2 dimer.

TDMP2 calculation of dynamic multipole polarizabilities and dispersion coefficients of the triplebonded molecules CO, N2, CN−, and NO+

In order to test the performance of the recently developed time-dependent second-order Mo/ller–Plesset perturbation theory (TDMP2) for anisotropic frequency-dependent multipole polarizabilities, we have studied the isoelectronic series CO, N2, CN−, and NO+. The polarizabilities of these triple-bonded diatomics are an interesting test for the TDMP2 method, because it is known that Mo/ller–Plesset perturbation theory has its difficulties describing multiple bonds. We selected these molecules as test systems because, especially for N2 and CO, accurate experimental and other ab initio data are available to compare with and their dynamical polarizabilities are needed for dispersion coefficients of van der Waals complexes, which are presently under intensive investigation. To get reliable results near the TDMP2 basis set limit we used large one-particle basis sets, optimized for polarizability calculations at the coupled Hartree–Fock level. The results show that the TDMP2 method is capable to improve for the isotropic as well as for the anisotropic polarizabilities considerably on the TDHF approximation, with the exception of the dipole polarizabilities of N2 and NO+, for which the static correlation effects are too strong to be treated by second-order Mo/ller–Plesset perturbation theory. However, we find, that the TDMP2 method, due to the use of coupled (TDHF) first-order orbital rotation parameters, is somewhat more stable with respect to static correlation effects than the singles-doubles-triples many-body perturbation theory to second order [SDT-MBPT(2)] double perturbation theory. Where reference data are available, the TDMP2 results for static polarizabilities and for the first Cauchy moments are in good agreement with the best theoretical and experimental data. We also calculated dispersion coefficients for the (N2)2 and the (CO)2 dimer and, utilizing the results of previous TDMP2 studies for the atoms He through Xe, also for the respective rare gas complexes of N2, CO, CN−, and NO+. We estimate the results to be the most accurate ab initio data available for these van der Waals coefficients.

The frequency dependence of nonlinear optical processes

Explicit formulas are derived for the sum rules for the frequency-dependent hyperpolarizability-diagonal-components. These are the counterparts to the Cauchy moments for the dynamic polarizabilities. The formulas allow for the frequency dependence of any nonlinear optical process to be expressed as a single general expansion up to terms which are of fourth power in the optical frequencies, Xnα,α,...,α(−ωσ;ω1,...,ωn)=Xn α,α,...,α(0)+AW2+BW22+B′W4, where ωσ=∑iωi, W2=ω2σ+ω21+...ω2n, and W4=ω4σ+ω41+...ω4n (in conventional notation X1=α, X2=β, X3=γ, etc.). The advantages of determining the frequency dependence of all NLO processes, for a given species, in a single calculation are stressed. We focus mainly on the sum rules (A, B, and B′) for X3 and X5. These are applicable to both atoms and molecules (with the exception of X5 for noncentrosymmetric molecules) and we evaluate them, using near-exact wave functions, for H and He. It is apparent that B′ is generally smaller than B and this accounts for the reasonable success of the Shelton–Bishop dispersion formula which is often used to fit experimentally-derived dynamic hyperpolarizabilities.

Topologically partitioned dynamic polarizabilities using the theory of atoms in molecules

Frequency-dependent distributed polarizabilities have been determined from time-dependent Hartree–Fock calculations, using the partitioning of the molecular space suggested by Bader's topological theory of atoms in molecules. The basis set dependence of the distributed dynamic polarizabilities is analyzed in terms of the first few Cauchy moments, for the carbon monoxide, water, cyanogen, urea and benzene molecules. Two alternative relocalization schemes have been considered in order to reduce the number of distributed dynamic polarizability parameters. The first one, closely related to the atomic polarizability model of Bader, leads to atomic charge–dipole and dipole–dipole polarizabilities, describing the response of the molecular charge distribution to a uniform external field, in terms of atomic charges and dipoles. The second scheme, similar to that suggested by Stone, retains the fully distributed description of the dynamic charge-flow polarizabilities, while all two-center dipole–dipole and charge–dipole contributions are condensed in one-center dynamic dipole–dipole polarizabilities. Key words: Bader-partitioning, distributed dynamic polarizabilities, Cauchy-moments, benzene, polarizability of; urea, polarizability of.

Dipole Polarizability, Cauchy Moments, and Related Properties of Hg

We report interferometric measurements of the frequency dependence of the refractive index n and the dipole polarizability α of Hg vapor. From the measured dispersion of the dipole polarizability the Cauchy moments S(−2) = 33.91(34), S(−4) = 279.8(8.4), and S(−6) = 14 100(2810) have been obtained. The static limit S(−2) is in good accordance with ab initio calculations and results obtained from dielectric constant measurements. The Cauchy moments are used to predict approximations of the dispersion interaction energy constant C6 = 255 au and the second hyperpolarizability γ(0) = 2.2 × 104 au.