Long-range, collision-induced hyperpolarizabilities of atoms or centrosymmetric linear molecules: Theory and numerical results for pairs containing H or He

Abstract

For atoms or molecules of D∞h or higher symmetry, this work gives equations for the long-range, collision-induced changes in the first (Δβ) and second (Δγ) hyperpolarizabilities, complete to order R−7 in the intermolecular separation R for Δβ, and order R−6 for Δγ. The results include nonlinear dipole-induced-dipole (DID) interactions, higher multipole induction, induction due to the nonuniformity of the local fields, back induction, and dispersion. For pairs containing H or He, we have used ab initio values of the static (hyper)polarizabilities to obtain numerical results for the induction terms in Δβ and Δγ. For dispersion effects, we have derived analytic results in the form of integrals of the dynamic (hyper)polarizabilities over imaginary frequencies, and we have evaluated these numerically for the pairs H...H, H...He, and He...He using the values of the fourth dipole hyperpolarizability ε(−iω; iω, 0, 0, 0, 0) obtained in this work, along with other hyperpolarizabilities calculated previously by Bishop and Pipin. For later numerical applications to molecular pairs, we have developed constant ratio approximations (CRA1 and CRA2) to estimate the dispersion effects in terms of static (hyper)polarizabilities and van der Waals energy or polarizability coefficients. Tests of the approximations against accurate results for the pairs H...H, H...He, and He...He show that the root mean square (rms) error in CRA1 is ∼20%–25% for Δβ and Δγ; for CRA2 the error in Δβ is similar, but the rms error in Δγ is less than 4%. At separations ∼1.0 a.u. outside the van der Waals minima of the pair potentials for H...H, H...He, and He...He, the nonlinear DID interactions make the dominant contributions to Δγzzzz (where z is the interatomic axis) and to Δγxxxx, accounting for ∼80%–123% of the total value. Contributions due to higher-multipole induction and the nonuniformity of the local field (Qα terms) may exceed 15%, while dispersion effects contribute ∼4%–9% of the total Δγzzzz and Δγxxxx. For Δγxxzz, the α term is roughly equal to the nonlinear DID term in absolute value, but opposite in sign. Other terms in Δγxxzz are smaller, but they are important in determining its net value because of the near cancellation of the two dominant terms. When Δγ is averaged isotropically over the orientations of the interatomic vector to give Δγ̄, dispersion effects dominate, contributing 76% of the total Δγ̄ (through order R−6) for H...H, 81% for H...He, and 73% for He...He.

Excess hyperpolarizabilities: the irreducible tensor approach

“…Excellent agreement with the results found by Buckingham et al [14] as well as the results by Hunt [23] and Li et al [12,13] has been found. It is interesting to note that the irreducible Cartesian components can be obtained directly from Cartesian considerations (see Sects.…”

“…It has been shown by Buckingham [14], Hunt [23] and Li et al [13], note also [15], that for a pair of atoms A and B the second order correction to the induced first-hyperpolarizability tensor β is given by the following formula (2) For the second order correction to the redundant induced second hyperpolarizability γ we have [14,13]:…”

“…. .∇ n−times 1 R , for microsystems separated by the distance R, play an important role [8][9][10][11][12][13][14][15]. If the described interactions result from the second order of the statistical perturbation theory, the relevant formulas typically contain the product of two such tensors [16].…”

Theoretical investigation of the ethylene dimer: Interaction energy and dipole moment

“…(9) by the expressions depending only on static polarizabilities. Thus, using constant ratio approximation, CRA2 [60] the first integral in eq. (9) can be written in the form…”

Collision-induced hyper-Rayleigh light scattering in gaseous dihydrogen-neon mixtures

“…The potential can be satisfactorily estimated by a series expansion of the form where the first term represents a short-range effect related to exchange or overlap type interactions [30,31], whereas the second part of the formula, distinguished from the series expansion, is introduced to enable the long-range part of the potential to diminish according to a ∼ 1/R 6 dependence; the other far-distance contributions (including dispersion-and induction-like processes) are entangled in the final summation. The q i parameters are obtained by fitting the function to the ab initio data sets reported in Ref.…”