Nonlocal Polarizability Densities Research Articles

Ion aggregation in high salt solutions. III. Computational vibrational spectroscopy of HDO in aqueous salt solutions.

The vibrational frequency, frequency fluctuation dynamics, and transition dipole moment of the O-D stretch mode of HDO molecule in aqueous solutions are strongly dependent on its local electrostatic environment and hydrogen-bond network structure. Therefore, the time-resolved vibrational spectroscopy the O-D stretch mode has been particularly used to investigate specific ion effects on water structure. Despite prolonged efforts to understand the interplay of O-D vibrational dynamics with local water hydrogen-bond network and ion aggregate structures in high salt solutions, still there exists a gap between theory and experiment due to a lack of quantitative model for accurately describing O-D stretch frequency in high salt solutions. To fill this gap, we have performed numerical simulations of Raman scattering and IR absorption spectra of the O-D stretch mode of HDO in highly concentrated NaCl and KSCN solutions and compared them with experimental results. Carrying out extensive quantum chemistry calculations on not only water clusters but also ion-water clusters, we first developed a distributed vibrational solvatochromic charge model for the O-D stretch mode in aqueous salt solutions. Furthermore, the non-Condon effect on the vibrational transition dipole moment of the O-D stretch mode was fully taken into consideration with the charge response kernel that is non-local polarizability density. From the fluctuating O-D stretch mode frequencies and transition dipole vectors obtained from the molecular dynamics simulations, the O-D stretch Raman scattering and IR absorption spectra of HDO in salt solutions could be calculated. The polarization effect on the transition dipole vector of the O-D stretch mode is shown to be important and the asymmetric line shapes of the O-D stretch Raman scattering and IR absorption spectra of HDO especially in highly concentrated NaCl and KSCN solutions are in quantitative agreement with experimental results. We anticipate that this computational approach will be of critical use in interpreting linear and nonlinear vibrational spectroscopies of HDO molecule that is considered as an excellent local probe for monitoring local electrostatic and hydrogen-bonding environment in not just salt but also other confined and crowded solutions.

Derivatives of the polarization propagator including orbital relaxation effects

In this article, we relate derivatives of the polarization propagator used in many-body theory to the nonlinear (quadratic) polarization propagator, and we relate derivatives of the quadratic polarization propagator to the nonlinear propagator of the next higher order, the cubic polarization propagator. We restrict the analysis to differentiation with respect to parameters eta for which the derivative of the Hamiltonian can be written as a sum of one-electron operators. Geometrical derivatives are obtained by specializing to the parameter eta to the alpha coordinate of nucleus I. We treat orbital relaxation explicitly by allowing for the eta dependence of creation and annihilation operators in the propagators. This treatment entails an extension of the geometrical derivative relations among response functions proven by Olsen and Jorgensen [J. Chem. Phys. 82, 3235 (1985)], because the propagator derivatives may involve changes in the one-electron orbitals that do not appear in the susceptibility derivatives. These results underlie the relations between Raman intensities and electric-field shielding tensors, which have been explained in terms of nonlocal polarizability and hyperpolarizability densities. The results suggest an alternative computational route to geometrical or other derivatives of both linear- and nonlinear-response functions: these derivatives can be evaluated without numerical differentiation, directly from the propagator of the next higher order.

Non-local dielectric functions on the nanoscale: electronic polarization and fluctuations

In this work, we use a non-local dielectric theory to describe the electronic polarization induced by perturbing fields, and to characterize spontaneous, quantum mechanical fluctuations in the polarization. The theory holds on the intramolecular length scale, within the Born-Oppenheimer approximation. A non-local dielectric function ε d ( r , r ′) acts as an integral kernel that gives the dielectric displacement D( r) at point r within a molecule, in terms of the electric field E( r′) acting at other points r ′. Due to the inhomogeneity of the intramolecular environment, ε d ( r , r ′) differs from the non-local dielectric function ε v ( r , r ′) that characterizes the screening of applied potentials; however, the longitudinal component of the spatial Fourier transform ε d ( k , k ′) is completely determined by k, k′, and ε v ( k , k ′). For a molecule in equilibrium with a radiation bath at a fixed temperature, the fluctuation-dissipation theorem relates correlations of the spontaneous polarization fluctuations to the imaginary part of the non-local polarizability density. From this result, we derive a relation between the correlations of the fluctuating electronic polarization and the dielectric function ε d ( r , r ′;iω).

Nonadditive, three-body dipoles and forces on nuclei: New interrelations and an electrostatic interpretation

This work establishes the relation between three-body forces and the polarization P(r), for a group of interacting molecules A...B...C with weak or negligible charge overlap. Using nonlocal polarizability and hyperpolarizability densities, we derive new equations for the three-body forces and dipoles due to dispersion and induction–dispersion interactions. For clusters containing at least one polar molecule, we identify a nonadditive induction–dispersion energy that was not included explicitly in previous analyses of van der Waals cluster spectra. Numerical estimates show that this effect is significant for Ar2...HCl and Ar2...DCl. Separately, we determine the effects of linear and nonlinear induction, dispersion, and induction–dispersion interactions on the polarization P(r); then we relate each contribution to P(r) to the corresponding component of the intermolecular force. Our work provides a classical electrostatic interpretation for all of the three-body forces at long range, including dispersion: The polarization of a molecular cluster produces these forces. This result holds even though particular interaction effects may generate a force but no net dipole, or a dipole but no net force.

Molecular softness, hypersoftness, infrared absorption, and vibrational Raman scattering: New relations derived from nonlocal polarizability densities

This work rigorously relates electromagnetic properties and characteristics of molecular potential energy surfaces to the empirical concept of ‘‘softness,’’ used to categorize Lewis acids and bases, and to summarize observed patterns of reactivity. New equations are derived that connect infrared absorption intensities, vibrational force constants, intermolecular forces at first order, and linear electric-field shielding tensors to softness kernels as defined in density functional theory. A generalization to nonlinear response—by introduction of the hypersoftness—leads to new equations in density-functional terms for vibrational Raman band intensities, the cubic anharmonicities in molecular potential energy surfaces, intermolecular forces at second order, and nonlinear electric-field shielding tensors. The analysis employs relations of the softness and hypersoftness to nonlocal polarizability and hyperpolarizability densities that represent the intramolecular distribution of response to inhomogeneous electric fields.

Vibrational force constants and anharmonicities: Relation to polarizability and hyperpolarizability densities

In this work, the derivatives of molecular potential energy surfaces V({R}) with respect to nuclear coordinates RK are related to derivatives of the electronic charge density with respect to applied electric fields. New equations are obtained for second, third, and fourth derivatives of V({R}) in terms of the charge density, the nonlocal polarizability density α(r,r′), and the hyperpolarizability densities β(r,r′,r″) and γ(r,r′,r″,r‴). In general, the nth derivative of the potential V({R}) depends on electrical susceptibility densities through (n−1)st order. The results hold for arbitrary nuclear coordinates {R}, not restricted to the equilibrium configuration {Re}. Specialization to {Re} leads to a new result for harmonic frequencies in terms of α(r,r′), and to new results for vibration–rotation coupling constants and anharmonicities in terms of α(r,r′), β(r,r′,r″) and higher-order hyperpolarizability densities. This work provides a simple physical interpretation for force derivatives obtained by use of analytic energy differentiation techniques in ab initio work, or in density functional theory: The charge reorganization terms in harmonic force constants give the electronic induction energy in the change of field δF due to an infinitesimal shift in nuclear positions. Cubic anharmonicity constants depend on the hyperpolarization energy of the electrons in the field δF, on the induction energy bilinear in δF and the second variation of the field δ2F, and on the gradients of the field from the unperturbed charge distribution. The results are derived by use of the Hohenberg–Kohn theorem or the electrostatic Hellmann–Feynman theorem, together with a chain of relations that connects the derivative of an electrical property of order n to the susceptibility density of order n+1. These derivatives are taken with respect to the nuclear coordinates RK, in contrast to the well known relations for derivatives with respect to an applied electric field. Analytic expressions are compared for the property derivatives that depend on susceptibility densities through γ(r,r′,r″,r‴). This includes the derivatives of V({R}) listed above; first, second, and third derivatives of the dipole moment; first and second derivatives of the polarizability; and the first derivative of the β hyperpolarizability with respect to the nuclear coordinates RK.

Force balance and force relay in molecular interactions: An analysis based on nonlocal polarizability densities

We have recently derived new results for dispersion, induction, and hyperpolarization forces, using nonlocal polarizability densities to characterize the changes in electronic charge density induced by molecular interactions. In this work, we prove that the fundamental physical requirement of force balance for two interacting molecules A and B is satisfied within the nonlocal response theory, order by order. An explicit proof is needed because of differences in the molecular properties that determine the forces on A and B. For example, at first order the force on A depends on the polarizability density of A, αA(r,r′;ω=0), while the first-order force on B depends on its polarizability density; and for distinct species A and B, there is no relation between αA(r,r′;ω=0) and αB(r,r′;ω=0). We show that force balance is derivable from a condition that we term ‘‘force relay.’’ Epstein has previously derived this condition for molecules in fixed external fields, assuming that the electronic state adjusts adiabatically to the perturbation: then the force of the external field on the nth order term in the electronic charge density equals the force on the nuclei due to the (n+1)st order correction to the electronic charge density. Our work generalizes the condition to external fields that are modified by and correlated with the changes in the electronic charge distribution, as for two interacting molecules with negligible charge overlap. Force relay is guaranteed by relations that we have established among permanent charge densities, linear response tensors, and nonlinear susceptibilities. All of the results stem from a hypervirial theorem applied to the electronic momentum operator, and hence from translational invariance. The results are not limited to the framework of the polarizability density theory, but also hold for the standard perturbation theory of interactions between nonoverlapping molecules, and for the Hellmann–Feynman theory of intermolecular forces.

Intramolecular screening of intermolecular forces

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.

Nonlocal polarizability density of a model system: A homogeneous electron gas at T=0

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 ε(k,0), and then use the connection to obtain results at three levels of approximation to ε(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.

Forces on nuclei in interacting molecules: New analytical results obtained with nonlocal polarizability densities

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.

Changes in electronic polarizability densities due to shifts in nuclear positions, and a new interpretation for integrated intensities of vibrational Raman bands

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.

Relationships between electric field shielding tensors and infrared or Raman intensities: An explanation based on nonlocal polarizability densities

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.

Field-induced fluctuation correlations and the effects of van der Waals interactions on molecular polarizabilities

New expressions for the van der Waals contribution to the collision-induced, static polarizability of a molecular pair (ΔαvdW) are derived within a reaction-field theory. For molecules interacting at long range, multipole expansions are used to determine the reaction field; at shorter range, where overlap is small but nonnegligible, the derivation is based on a nonlocal polarizability density model. In both cases, we obtain ΔαvdW in terms of integrals over imaginary frequencies, each involving the product of a hyperpolarizability for one molecule and a polarizability or hyperpolarizability for the other molecule. In addition, we show that the polarizability changes induced by van der Waals interactions between two molecules stem from two distinct physical effects. First, in an applied field F, each molecule is polarized nonlinearly by the simultaneous action of the field due to the fluctuating charge distribution of its neighbor and the field F. Second, the applied field F alters the correlations between the spontaneous, quantum mechanical fluctuations in the charge density of each molecule, thus affecting its interaction with the neighboring molecule. Effects of field-induced fluctuation correlations have not been included in earlier models for the van der Waals contributions to pair polarizabilities.

Nonlocal polarizability densities and the effects of short-range interactions on molecular dipoles, quadrupoles, and polarizabilities

Light scattering and absorption processes that are single-molecule forbidden may be observed in compressed gases and liquids, because of interaction-induced distortions of molecular charge distributions. In order to extract information about intermolecular dynamics from line shape studies of ‘‘forbidden’’ spectroscopic processes, it is necessary to determine collision-induced electrical properties as functions of the separations and relative orientations of the interacting molecules. When overlap is not negligible and series expansions valid at long range break down, a model that represents the distribution of polarizable matter in the interacting molecules by means of nonlocal polarizability densities is applicable in computing these properties. In the model the field that induces nonlocal polarization in one molecule is the sum of an external field and the field due to the polarization (permanent, fluctuating, and induced) of the collision partner. With the inclusion of nonlinear response, the model yields accurate expressions for the induction and dispersion contributions to collision-induced dipoles and polarizabilities; exchange effects are additive to lowest order. To first order in the atomic interaction, analytical results are presented for the dipole induced in one hydrogen atom by the approach of a second, the collision-induced pair quadrupole, and the pair polarizability tensor.

Nonlocal polarizability densities and van der Waals interactions

The van der Waals interaction energy of a molecular pair is derived within a simple physical model that represents the distribution of polarizable matter in each molecule by the use of nonlocal polarizability densities. In the model the field due to the instantaneous polarization of one molecule polarizes the second molecule nonlocally, thereby producing a reaction field that acts on the first molecule. The resulting energy change depends upon the averaged product of the fluctuating polarization at different points in the first molecule; by the fluctuation–dissipation theorem, this product is related to the imaginary part of the nonlocal polarizability density. The model gives the van der Waals energy as an integral of a contracted tensor product of two dipole propagators and the imaginary-frequency nonlocal polarizabilities of the interacting molecules. The validity of the model is established by comparison with results from second-order perturbation theory.