ALBERT

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: 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: 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: 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: The nonlocal polarizability density α(r,r';ω) gives the polarization induced at a point r in a quantum mechanical system, due to a perturbing field of frequency ω that acts at the point r', within linear response; thus it reflects the distribution of polarizability in the system. In order to gain information about the nature and functional form of α(r,r';ω), in this work we analyze the nonlocal polarizability density of a well-characterized system, a homogeneous electron gas at zero temperature. We establish a connection between the static, longitudinal component of the nonlocal polarizability density in position space and the dielectric function cursive-epsilon(k,0), and then use the connection to obtain results at three levels of approximation to cursive-epsilon(k,0): We compare the Thomas–Fermi (TF), random phase approximation (RPA), and Vashishta–Singwi (VS) forms. At TF level, we evaluate the nonlocal polarizability density analytically, while within the RPA we obtain asymptotic analytical results. The RPA and VS results are similar, and qualitatively distinct from the TF results, which diverge as ||r−r'|| approaches zero. Within the RPA, we find two long-range components in αL(r,r';0): The first is a monotonically decreasing component that arises from charge screening in the electron gas, and varies as ||r−r'||−3; the second is an oscillatory component with terms of order ||r−r'||−n (n≥3) associated with Friedel oscillations in the electron density. These results indicate the possibility of long-range, intramolecular terms in the nonlocal polarizability densities of individual molecules.

Notes: By use of nonlocal polarizability densities, we analyze the intramolecular screening of intermolecular fields. For two interacting molecules A and B with weak or negligible charge overlap, we show that the reaction field and the field due to the unperturbed charge distribution of the neighboring molecule are screened identically via the Sternheimer shielding tensor and its generalizations to nonuniform fields and nonlinear response. The induction force on nucleus I in molecule A, derived from perturbation theory, results from linear screening of the reaction field due to B and nonlinear screening of the field from the permanent charge distribution of B. In general, at first or second order in the molecular interaction, the screening-tensor expressions for the force on nucleus I involve susceptibilities of one order higher than the expressions derived from perturbation theory. The first-order force from perturbation theory involves permanent charge moments, while the first-order screened force involves linear response tensors; and the second-order screened force depends on hyperpolarizabilities, while second-order induction effects are specified in terms of static, lowest-order susceptibilities. The equivalence of the two formulations for these forces, order by order, is a new illustration of the interrelations we have found among permanent moments, linear-response tensors, and nonlinear response. This work also provides new insight into the dispersion forces on an individual nucleus I in molecule A by separating the forces into two distinct terms—the first term results from changes in the reaction of A to the fluctuating charge distribution of the neighboring molecule B, when nucleus I shifts infinitesimally, and the second term stems from changes in correlations of the fluctuating charge distribution of A itself. Changes in the fluctuation correlations are determined by changes in the classical Coulomb field of nucleus I and by the imaginary part of the hyperpolarizability density of A. The full dispersion force on nucleus I in A is equivalent to the screened force of an effective fluctuating field due to B at imaginary frequencies.

Notes: The thermodynamic and stochastic theory of chemical systems far from equilibrium is extended to reactions in inhomogeneous system for both single and multiple intermediates, with multiple stationary states coupled with linear diffusion. The theory is applied to the two variable Selkov model coupled with diffusion, in particular to the issue of relative stability of two stable homogeneous stationary states as tested in a possible inhomogeneous experimental configuration. The thermodynamic theory predicts equistability of such states when the excess work from one stationary state to the stable inhomogeneous concentration profile equals the excess work from the other stable stationary state. The predictions of the theory on the conditions for relative stability are consistent with solutions of the deterministic reaction-diffusion equations. In the following article we apply the theory again to the issue of relative stability for single-variable systems, and make comparison with numerical solutions of the reaction-diffusion equations for the Schlögl model, and with experiments on an optically bistable system where the kinetic variable is temperature and the transport mechanism is thermal conduction.