ALBERT

Notes: When the charge overlap between interacting molecules or ions A and B is weak or negligible, the first-order interaction energy depends only upon the molecular positions, orientations, and the unperturbed charge distributions of the molecules. In contrast, the first-order force on a nucleus in molecule A as computed from the Hellmann–Feynman theorem depends not only on the unperturbed charge distribution of molecule B, but also on the electronic polarization induced in A by the field from B. At second order, the interaction energy depends on the first-order, linear response of each molecule to its neighbor, while the Hellmann–Feynman force on a nucleus in A depends on second-order and nonlinear responses to B. One purpose of this work is to unify the physical interpretations of interaction energies and Hellmann–Feynman forces at each order, using nonlocal polarizability densities and connections that we have recently established among permanent moments, linear response, and nonlinear response tensors. Our theory also yields new information on the origin of terms in the long-range forces on molecules, through second order in the interaction.One set of terms in the force on molecule A is produced by the field due to the unperturbed charge distribution of B and by the static reaction field from B, acting on the nuclear moments of A. This set originates in the direct interactions between the nuclei in A and the charge distribution of B. A second set of terms results from the permanent field and the reaction field of B acting on the permanent electronic moments of A. This set results from the attraction of nuclei in A to the electronic charge in A itself, polarized by linear response to B. Finally, there are terms in the force on A due to the perturbation of B by the static reaction field from A; these terms stem from the attraction of nuclei in A to the electronic charge in A, hyperpolarized by the field from B. For neutral, dipolar molecules A and B at long range, the forces on individual nuclei vary as R−3 in the intermolecular separation R; but when the forces are summed over all of the nuclei, the vector sum varies as R−4. This result, an analogous conversion at second order (from R−6 forces on individual nuclei to an R−7 force when summed over the nuclei), and the long-range limiting forces on ions are all derived from new sum rules obtained in this work.

Notes: New equations for the derivatives of molecular dipole moments and polarizabilities with respect to nuclear coordinates are derived in terms of nonlocal polarizability densities, linear and nonlinear. New equations are also derived for the electric field shielding tensors at nuclei of molecules in static external fields of arbitrary spatial variation. Both involve integrals of the dipole propagator and the polarizability densities. This analysis explains the relationship between the linear electric field shielding tensors and the infrared intensity for a vibrational mode; it also accounts for the relationship between the quadratic electric field shielding tensors and the Raman intensity, as well as relations connecting higher-order shielding tensors to hyper-Raman intensities. When a nucleus moves infinitesimally, the electronic charge distribution responds via its nonlocal polarizability density to the change in the Coulomb field due to that nucleus, and this produces the change in the electronic dipole moment. All of the quantum mechanical effects are contained within the polarizability density. Analogously, the change in the Coulomb field and response via the hyperpolarizability densities determine the change in electronic polarizability when a nuclear position shifts.

Notes: Transient changes in polarizability during collisions between atoms and molecules give rise to interaction-induced rototranslational Raman scattering: the scalar component of the collision-induced polarizability Δα00 accounts for isotropic scattering, while the second-rank component ΔαM2 accounts for collision-induced depolarized scattering. We have evaluated the changes in electronic polarizability due to interactions between an atom and a molecule of D∞h symmetry in fixed configurations, with nonoverlapping charge distributions. We have cast the resulting expressions into the symmetry-adapted form used in spectroscopic line shape analyses. Our results are complete to order R−6 in the atom–molecule separation R. To this order, the collision-induced change in polarizability of an atom and a D∞h molecule reflects not only dipole-induced–dipole (DID) interactions, but also molecular polarization due to the nonuniformity of the local field, polarization of the atom in the field due to higher multipoles induced in the molecule, hyperpolarization of the atom by the applied field and the quadrupolar field of the molecule, and dispersion. We have analyzed the dispersion contributions to the atom–molecule polarizability within our reaction-field model, which yields accurate integral expressions for the polarizability coefficients. For numerical work, we have also developed approximations in terms of static polarizabilities, γ hyperpolarizabilities, and dispersion energy coefficients. Estimated polarizability coefficients are tabulated for H, He, Ne, and Ar atoms interacting with H2 or N2 molecules. The mean change in polarizability Δα¯, averaged over the orientations of the molecular axis and the vector between atomic and molecular centers, is determined by second-order DID interactions and dispersion. For the lighter pairs, dispersion terms are larger than second-order DID terms in Δα¯. In both Δα00 and ΔαM2, first-order DID interactions dominate at long range; other interaction effects are smaller, but detectable. At long range, the largest deviations from the first-order DID results for Δα00 areproduced by dispersion terms for lighter species considered here and by second-order DID terms for the heavier species; in ΔαM2, the largest deviations from first-order DID results stem from the effects of field nonuniformity and higher multipole induction, for atoms interacting with N2.

Notes: The nonlocal polarizability density α(r;r',ω) is a linear-response tensor that determines the electronic polarization induced at point r in a molecule, by an external electric field of frequency ω, acting at r'. This work focuses on the change in α(r;r',ω) when a nuclear position shifts infinitesimally. We prove directly that the electronic charge distribution responds to the change in Coulomb field due to the nucleus via the same hyperpolarizability density that describes its response to external fields. This generalizes a result found previously for the static (ω=0) polarizability density. The work also provides a new interpretation for the integrated intensities of vibrational Raman bands: it proves that the intensities depend on the hyperpolarizability densities and the dipole propagator.

Notes: This paper provides the first explicit, general proof that the leading-order dispersion forces between two interacting molecules result from the attraction of nuclei in each molecule to the dispersion-induced change in the electronic charge density of the same molecule. The proof given here holds for molecules of any symmetry, provided that overlap between the charge distributions is small. New sum rules for the nonlinear response tensors are also obtained, after consideration of the long-range limit. A perturbation analysis gives the dispersion-induced polarization in each molecule in terms of nonlocal, nonlinear response tensors taken at imaginary frequencies. Forces on the nuclei are computed from a reaction-field expression for the dispersion energy, in terms of polarizability densities. Recent work has shown that the derivative of the polarizability density with respect to a nuclear coordinate is linked to an integral involving the nonlinear response tensor and the dipole propagator, and this link provides the key to the proof.

Notes: Using a reaction field model, we have derived symmetry-adapted expressions for the van der Waals dipoles of atom–heteroatom and atom-D∞h molecule pairs, with nonoverlapping charge distributions.The leading van der Waals contributions vary as R−7 in the intermolecular separation R and depend upon products of the polarizability ααβ(iω) of one molecule with a dipole–quadrupole hyperpolarizability Bαβ,γδ (0,iω) of the other, integrated over imaginary frequencies. We have developed new approximations for these integrals in terms of the static polarizabilities ααβ, the hyperpolarizabilities Bαβ,γδ, and the van der Waals energy coefficients C6 and C8 (both isotropic and anisotropic components for atom–molecule pairs). The approximations agree well with accurate perturbation results for two model systems. Applied to He⋅⋅⋅H2, our approximations give the first direct results for the leading van der Waals contributions to the dipole. In two symmetry components of the He⋅⋅⋅H2 dipole at long range, van der Waals effects are larger than induction effects; both should be included in fitting collision-induced roto-translational spectra.

Notes: We have derived symmetry-adapted expressions for the dipole moments of pairs of D∞h molecules interacting at long range, in a form useful for line shape analyses of collision-induced rototranslational spectra. Our results are complete to order R−7 in the intermolecular separation R. In addition to quadrupolar and hexadecapolar induction effects, results to this order include induction due to nonuniformities in the local field acting on a molecule (E-tensor induction), back induction, and polarization due to dispersion forces. The dispersion terms are computed within our recently developed reaction field model, from which we have obtained accurate integral expressions for the dipole coefficients, and approximations in terms of static susceptibilities and dispersion energy coefficients. For H2⋅⋅⋅H2, H2⋅⋅⋅N2, and N2⋅⋅⋅N2, numerical results for the dipole coefficients are tabulated. While quadrupolar induction dominates the long-range dipole, other induction effects are evident in the far-infrared collision-induced spectra. Over the range of validity of the model, E-tensor induction, back induction, and dispersion effects are generally smaller than hexadecapolar induction, but appreciable. The magnitudes of the dipole coefficients and the orientation dependence of each polarization mechanism determine its contributions to the observed collision-induced absorption spectra.

Notes: One contribution to the transient dipoles induced during molecular collisions comes from the van der Waals interactions between the fluctuating charge distributions of the colliding molecules. In this paper we show that the application of an external field to a molecular pair changes the van der Waals interaction energy in two ways. First, the field alters the response of each molecule to the nonuniform, fluctuating field of its neighbor. Second, the applied field induces new correlations between the fluctuating charge moments on each molecular center. Field-induced fluctuation correlations have not been included in earlier models for the van der Waals dipole. With the inclusion of these effects, the van der Waals dipole of a molecular pair is simply related to an imaginary-frequency integral of two or more terms, each involving the product of a linear response tensor for one molecule and a nonlinear response tensor for the neighboring molecule.

Notes: Collision-induced light scattering spectra of the inert gases and hydrogen at high densities provide evidence of nonadditive three-body interaction effects, for which a quantitative theory is needed. In this work, we derive and evaluate the three-body polarizability Δα(3) for interacting molecules with negligible electronic overlap. Our results, based on nonlocal response theory, account for dipole-induced-dipole (DID) interactions, quadrupolar induction, dispersion, and concerted induction-dispersion effects. The contribution of leading order comes from a DID term that scales as α3d−6 in the molecular polarizability α and a representative distance d between the molecules in a cluster. Quadrupolar induction effects are also large, however, ranging from ∼35% to 104% of the leading DID terms for equilateral triangular configurations of the species studied in this work, at separations approximately 1 a.u. beyond the van der Waals minima in the isotropic pair potentials. For the same configurations, the dispersion terms range from 2% to 7% of the total Δα¯(3). The dispersion and induction-dispersion contributions are derived analytically in terms of integrals over imaginary frequency, with integrands containing the polarizability α(iω) and the γ hyperpolarizability. For H, He, and H2, the integrals have been evaluated accurately by 64-point Gauss–Legendre quadrature; for heavier species, we have developed approximations in terms of static polarizabilities, static hyperpolarizabilities, and van der Waals interaction energy coefficients (C6 and C9). In the isotropic interaction-induced polarizability Δα¯, the three-body terms are comparable in magnitude to the two-body terms, due to a cancellation of the first-order, two-body DID contributions to Δα¯. For the heavier species in this work (Ar, Kr, Xe, N2, CH4, and CO2) in the configurations studied, the three-body contributions to Δα¯ range from −7 to −9% of the two-body terms for equilateral triangular arrays and from 35% to 47% of the two-body terms for linear, centrosymmetric systems. © 2000 American Institute of Physics.

Notes: Band intensities for nonresonant vibrational hyper-Raman scattering depend on the derivatives of the β hyperpolarizability, a nonlinear electronic response tensor, with respect to normal mode coordinates. In this work, we derive a new result for the change in β(−ωσ; ω1,ω2) due to small shifts in nuclear positions within a molecule. We prove that the derivative of β(−ωσ; ω1,ω2), taken with respect to the position RK of nucleus K, depends on the nonlocal hyperpolarizability density γ(r,r′,r″,r(triple-prime); −ωσ; ω1,ω2,0) of second order, the charge on nucleus K, and the dipole propagator from RK to r(triple-prime). Thus γ(r,r′,r″,r(triple-prime); −ωσ; ω1,ω2,0) determines the origins of vibrational hyper-Raman intensities on the intramolecular scale. Two observations provide the physical basis for this result: The effective value of β for a molecule in a static applied field is governed by the γ hyperpolarizability density. When a nucleus shifts infinitesimally, the electrons respond to the resulting change in the nuclear Coulomb field via the same nonlocal susceptibilities that characterize their response to an applied electric field. © 1995 American Institute of Physics.