L. Molecular refractivity and atomic interaction. II

Development of discrete interaction models for ultra-fine nanoparticle plasmonics

Plasmonics serves as a most outstanding feature of nanoparticle technology and is nowadays used in numerous applications within imaging, sensing and energy harvesting, like plasmonically enhanced solar cells, nanoparticle bioimaging, plasmon-controlled fluorescence for molecular tracking in living cells, plasmon-controlled electronic molecular devices and surface enhanced Raman spectroscopy for single molecular detection. Although plasmonics has been utilized since ancient times, the understanding of its basic interactions has not been fully achieved even under the emergence of modern nanoscience. In particular, it has been difficult to address the "ultra-fine" 1-10 nm regime, important for applications especially in bioimaging and biomedical areas, where neither classical nor quantum based theoretical methods apply. Recently, new approaches have been put forward to bridge this size gap based on semi-empirical discrete interaction models where each atom makes a difference. A primary aim of this perspective article is to review some of the most salient features of these models, and in particular focus on a recent extension - the extended discrete interaction model (Ex-DIM), where the geometric and environmental features are extended - and highlight a set of benchmark studies using this model concerning size, shape, material, temperature dependence and other characteristics of ultra-fine plasmonic nanoparticles. We also analyze new possibilities offered by the model for designing ultra-fine plasmonic particles for applications in the areas of bioimaging, biosensing, photothermal therapy, infrared light harvesting and photodetection. We foresee that future modelling activities will be closely connected to collaborative experimental work including synthesis, device fabrication and measurements with feedback and validation in a systematic fashion. With this strategy we can expect that modelling of ultra-fine plasmonics particles can be integrated in the development of novel plasmonic systems with unprecedented performance and applicability.

A recursive cell multipole method for atomistic electrodynamics models

For large plasmonic nanoparticles, retardation effects become important once their length becomes comparable to the wavelength of light. However, most models do not incorporate retardation effects due to the high computational cost of solving for the optical properties of large atomistic electrodynamics systems. In this work, we derive and implement a recursive fast multipole method (FMM) in Cartesian coordinates that includes retardation effects. In this method, higher-order electrodynamic interaction tensors used for the FMM are calculated recursively, thus greatly reducing the implementation complexity of the model. This method allows for solving of the optical properties of large atomistic nanoparticles with controlled accuracy; in practice, taking the expansion to the fifth order provides a good balance of accuracy and computational time. Finally, we study the effects retardation has on the near- and far-field properties of large plasmonic nanoparticles with over a million atoms using this method. We specifically focus on nanorods and their dimers, which are known to generate highly confined fields in their junctions. In the future, this method can be applied to simulations in which accurate near-field properties are required, such as surface-enhanced Raman scattering.

Modeling the near-field effect on molecular excited states using the discrete interaction model/quantum mechanical method

Strong light-matter interactions significantly modify the optical properties of molecules in the vicinity of plasmonic metal nanoparticles. Since the dimension of the plasmonic cavity approaches that of the molecules, it is critical to explicitly describe the nanoparticle junctions. In this work, we use the discrete interaction model/quantum mechanical (DIM/QM) method to model the coupling between the plasmonic near-field and molecular excited states. DIM/QM is a combined electrodynamics/quantum mechanical model that uses an atomistic description of the nanoparticle. We extend the DIM/QM method to include the local field effects in the sum-over-state formalism of time-dependent density functional theory. As a test of the method, we study the interactions between small organic chromophores and metal nanoparticles. In particular, we examine how the inclusion of multiple electronic transitions and intermolecular interactions modify the coupling between molecules and nanoparticles. Using the sum-over-state formalism of DIM/QM, we show that two-state models break down when the plasmon excitation is detuned from the molecular excitations. To gain further insight, we compare the simple coupled-dipole model (CDM) with the DIM/QM model. We find that CDM works well for simple systems but fails when going beyond the single molecule or single nanoparticle cases. We also find that the coupling depends strongly on the site of the nanoparticle in which the chromophore couples to. Our work suggests the importance of explicitly describing the cavity to capture the atomistic level local field environment in which the molecule strongly couples to.

Molecular Interactions Induced by a Static Electric Field in Quantum Mechanics and Quantum Electrodynamics

By means of quantum mechanics and quantum electrodynamics applied to coupled harmonic Drude oscillators, we study the interaction between two neutral atoms or molecules subject to a uniform static electric field. Our focus is to understand the interplay between leading contributions to field-induced electrostatics/polarization and dispersion interactions, as considered within the employed Drude model for both non-retarded and retarded regimes. For the first case, we present an exact solution for two coupled oscillators obtained by diagonalizing the corresponding quantum-mechanical Hamiltonian and demonstrate that the external field can control the strength of different intermolecular interactions and relative orientations of the molecules. In the retarded regime described by quantum electrodynamics, our analysis shows that field-induced electrostatic and polarization energies remain unchanged (in isotropic and homogeneous vacuum) compared to the non-retarded case. For interacting species modeled by quantum Drude oscillators, the developed framework based on quantum mechanics and quantum electrodynamics yields the leading contributions to molecular interactions under the combined action of external and vacuum fields.

Decomposition of molecular properties

We review recent work on property decomposition techniques using quantum chemical methods and discuss some topical applications in terms of quantum mechanics-molecular mechanics calculations and the constructing of properties of large molecules and clusters. Starting out from the so-called LoProp decomposition scheme [Gagliardi et al., J. Chem. Phys., 2004, 121, 4994] for extracting atomic and inter-atomic contributions to molecular properties we show how this method can be generalized to localized frequency-dependent polarizabilities, to localized hyperpolarizabilities and to localized dispersion coefficients. Some applications of the generalized decomposition technique are reviewed - calculations of frequency-dependent polarizabilities, Rayleigh scattering of large clusters, and calculations of hyperpolarizabilities of proteins.

Plasmonic Surface Lattice Resonances: A Review of Properties and Applications

When metal nanoparticles are arranged in an ordered array, they may scatter light to produce diffracted waves. If one of the diffracted waves then propagates in the plane of the array, it may couple the localized plasmon resonances associated with individual nanoparticles together, leading to an exciting phenomenon, the drastic narrowing of plasmon resonances, down to 1–2 nm in spectral width. This presents a dramatic improvement compared to a typical single particle resonance line width of >80 nm. The very high quality factors of these diffractively coupled plasmon resonances, often referred to as plasmonic surface lattice resonances, and related effects have made this topic a very active and exciting field for fundamental research, and increasingly, these resonances have been investigated for their potential in the development of practical devices for communications, optoelectronics, photovoltaics, data storage, biosensing, and other applications. In the present review article, we describe the basic physical principles and properties of plasmonic surface lattice resonances: the width and quality of the resonances, singularities of the light phase, electric field enhancement, etc. We pay special attention to the conditions of their excitation in different experimental architectures by considering the following: in-plane and out-of-plane polarizations of the incident light, symmetric and asymmetric optical (refractive index) environments, the presence of substrate conductivity, and the presence of an active or magnetic medium. Finally, we review recent progress in applications of plasmonic surface lattice resonances in various fields.

An empirical extrapolation scheme for efficient treatment of induced dipoles

Many cutting edge force fields include polarization, to enhance their accuracy and range of applicability. In this work, we develop efficient strategies for the induced dipole polarization method. By fitting various orders of perturbation theory (PT) dipoles to a diverse training set, we arrive at a family of fully analytic methods - whose nth order is referred to OPTn - that span the full spectrum of polarization methods from the fast zeroth-order approach that neglects mutual dipole coupling, approaching the fully variational approach at high order. Our training set contains many difficult cases where the PT series diverges, and we demonstrate that our OPTn methods still deliver excellent results in these cases. Our tests show that the OPTn methods exhibit rapid convergence towards the exact answer with each increasing PT order. The fourth order OPT4 method, whose costs are commensurate with three iterations of the leading conjugate gradient method, is a particularly promising candidate to be used as a drop-in replacement for existing solvers without further parameterization.

Polarizable Embedding Density Matrix Renormalization Group

The polarizable embedding (PE) approach is a flexible embedding model where a preselected region out of a larger system is described quantum mechanically, while the interaction with the surrounding environment is modeled through an effective operator. This effective operator represents the environment by atom-centered multipoles and polarizabilities derived from quantum mechanical calculations on (fragments of) the environment. Thereby, the polarization of the environment is explicitly accounted for. Here, we present the coupling of the PE approach with the density matrix renormalization group (DMRG). This PE-DMRG method is particularly suitable for embedded subsystems that feature a dense manifold of frontier orbitals which requires large active spaces. Recovering such static electron-correlation effects in multiconfigurational electronic structure problems, while accounting for both electrostatics and polarization of a surrounding environment, allows us to describe strongly correlated electronic structures in complex molecular environments. We investigate various embedding potentials for the well-studied first excited state of water with active spaces that correspond to a full configuration-interaction treatment. Moreover, we study the environment effect on the first excited state of a retinylidene Schiff base within a channelrhodopsin protein. For this system, we also investigate the effect of dynamical correlation included through short-range density functional theory.

First Hyperpolarizability of Collagen Using the Point Dipole Approximation

The application of localized hyperpolarizabilities to predict a total protein hyperpolarizability is presented for the first time, using rat-tail collagen as a demonstration example. We employ a model comprising the quadratic Applequist point-dipole approach, the so-called LoProp transformation, and a procedure with molecular fractionation using conjugate caps to determine the atomic and bond contributions to the net β tensor of the collagen [(PPG)10]3 triple-helix. By using Tholes exponential damping modification to the dyadic tensor in the Applequist equations, a correct qualitative agreement with experiment is found. The intensity of the βHRS signal and the depolarization ratios are best reproduced by decomposing the LoProp properties into the atomic positions and using Tholes exponential damping with the original damping parameter. Some ramifications of the model for general protein property optimization are briefly discussed.

Efficient treatment of induced dipoles

Most existing treatments of induced dipoles in polarizable molecular mechanics force field calculations use either the self-consistent variational method, which is solved iteratively, or the "direct" approximation that is non-iterative as a result of neglecting coupling between induced dipoles. The variational method is usually implemented using assumptions that are only strictly valid under tight convergence of the induced dipoles, which can be computationally demanding to enforce. In this work, we discuss the nature of the errors that result from insufficient convergence and suggest a strategy that avoids such problems. Using perturbation theory to reintroduce the mutual coupling into the direct algorithm, we present a computationally efficient method that combines the precision of the direct approach with the accuracy of the variational approach. By analyzing the convergence of this perturbation series, we derive a simple extrapolation formula that delivers a very accurate approximation to the infinite order solution at the cost of only a few iterations. We refer to the new method as extrapolated perturbation theory. Finally, we draw connections to our previously published permanent multipole algorithm to develop an efficient implementation of the electric field and Thole terms and also derive some necessary, but not sufficient, criteria that force field parameters must obey.

Theory of Linear and Nonlinear Surface-Enhanced Vibrational Spectroscopies

The vibrational spectroscopy of molecules adsorbed on metal nanoparticles can be enhanced by many orders of magnitude so that the detection and identification of single molecules are possible. The enhancement of most linear and nonlinear vibrational spectroscopies has been demonstrated. In this review, we discuss theoretical approaches to understanding linear and nonlinear surface-enhanced vibrational spectroscopies. A unified description of enhancement mechanisms classified as either electromagnetic or chemical in nature is presented. Emphasis is placed on understanding the spectral changes necessary for interpretation of linear and nonlinear surface-enhanced vibrational spectroscopies.

Hyperpolarizabilities of extended molecular mechanical systems

The equations for decomposition of the molecular hyperpolarizabilty into atomic sites is derived alongside with equations to calculate the cluster hyperpolarizability as resulting from the second order induced dipole moments in the point-dipole model.

Frequency-Dependent Polarizabilities of Amino Acids as Calculated by an Electrostatic Interaction Model

The frequency-dependent polarizability of the 20 essential amino acids has been calculated by an electrostatic interaction model where an Unsöld-type of model has been adopted for the frequency dependence. The interaction model has previously been parametrized from Hartree-Fock calculations on a set of molecules, and the model is in this work extended by sulfur parameters by including a set of 18 small sulfur compounds. The results for the amino acids by using the interaction model compare well with Hartree-Fock calculations with deviations of around 5% for the isotropic polarizability. Furthermore, the intrinsic (or optical) dielectric constant related to the polarizability has been calculated for three small proteins, ribonuclease inhibitor, lysozyme, and green fluorescent protein, adopting the interaction model. The results are consistent with the intrinsic dielectric constants found for proteins in the literature.

On combining Thole's induced point dipole model with fixed charge distributions in molecular mechanics force fields

The Thole induced point dipole model is combined with three different point charge fitting methods, Merz-Kollman (MK), charges from electrostatic potentials using a grid (CHELPG), and restrained electrostatic potential (RESP), and two multipole algorithms, distributed multipole analysis (DMA) and Gaussian multipole model (GMM), which can be used to describe the electrostatic potential (ESP) around molecules in molecular mechanics force fields. This is done to study how the different methods perform when intramolecular polarizability contributions are self-consistently removed from the fitting done in the force field parametrization. It is demonstrated that the polarizable versions of the partial charge models provide a good compromise between accuracy and computational efficiency in describing the ESP of small organic molecules undergoing conformational changes. For the point charge models, the inclusion of polarizability reduced the the average root mean square error of ESP over the test set by 4-10%.

Local electric field factors by a combined charge-transfer and point–dipole interaction model

A model for the local electric field as a linear response to a frequency-dependent external electric field is presented based on a combined charge-transfer and point–dipole interaction force-field model.

Hydrothermal properties of the COS/D2 water model: a polarizable charge-on-spring water model, at elevated temperatures and pressures

Molecular simulations have been conducted to assess the pVT properties and static permittivity of the charge-on-spring polarizable water model COS/D2 at hydrothermal conditions from 300 to 450 °C and bulk densities of 0.001 to 1.0 g cm⁻³.

Multiproperty empirical isotropic interatomic potentials for CH4-inert gas mixtures

An approximate empirical isotropic interatomic potentials for CH₄–inert gas mixtures are developed by simultaneously fitting the Exponential-Spline-Morse-Spline-van der Waals (ESMSV) potential form to viscosity, thermal conductivity, thermal diffusion factors, diffusion coefficient, interaction second pressure virial coefficient and scattering cross-section data. Quantum mechanical lineshapes of collision-induced absorption (CIA) at different temperatures for CH₄–He and at T = 87 K for CH₄–Ar are computed using theoretical values for overlap, octopole and hexadecapole mechanisms and interaction potential as input. Also, the quantum mechanical lineshapes of collision-induced light scattering (CILS) for the mixtures CH₄–Ar and CH₄–Xe at room temperature are calculated. The spectra of scattering consist essentially of an intense, purely translational component which includes scattering due to free pairs and bound dimers, and the other is due to the induced rotational scattering. These spectra have been interpreted by means of pair-polarizability terms, which arise from a long-range dipole-induced-dipole (DID) with small dispersion corrections and a short-range interaction mechanism involving higher-order dipole–quadrupole A and dipole–octopole E multipole polarizabilities. Good agreement between computed and experimental lineshapes of both absorption and scattering is obtained when the models of potential, interaction-induced dipole and polarizability components are used.

Nonmetallic electronegativity equalization and point-dipole interaction model including exchange interactions for molecular dipole moments and polarizabilities

The electronegativity equalization model (EEM) has been combined with a point-dipole interaction model to obtain a molecular mechanics model consisting of atomic charges, atomic dipole moments, and two-atom relay tensors to describe molecular dipole moments and molecular dipole-dipole polarizabilities. The EEM has been phrased as an atom-atom charge-transfer model allowing for a modification of the charge-transfer terms to avoid that the polarizability approaches infinity for two particles at infinite distance and for long chains. In the present work, these shortcomings have been resolved by adding an energy term for transporting charges through individual atoms. A Gaussian distribution is adopted for the atomic charge distributions, resulting in a damping of the electrostatic interactions at short distances. Assuming that an interatomic exchange term may be described as the overlap between two electronic charge distributions, the EEM has also been extended by a short-range exchange term. The result is a molecular mechanics model where the difference of charge transfer in insulating and metallic systems is modeled regarding the difference in bond length between different types of system. For example, the model is capable of modeling charge transfer in both alkanes and alkenes with alternating double bonds with the same set of carbon parameters only relying on the difference in bond length between carbon sigma- and pi-bonds. Analytical results have been obtained for the polarizability of a long linear chain. These results show that the model is capable of describing the polarizability scaling both linearly and nonlinearly with the size of the system. Similarly, a linear chain with an end atom with a high electronegativity has been analyzed analytically. The dipole moment of this model system can either be independent of the length or increase linearly with the length of the chain. In addition, the model has been parametrized for alkane and alkene chains with data from density functional theory calculations, where the polarizability behaves differently with the chain length. For the molecular dipole moment, the same two systems have been studied with an aldehyde end group. Both the molecular polarizability and the dipole moment are well described as a function of the chain length for both alkane and alkene chains demonstrating the power of the presented model.

Enhanced polarizability of aromatic molecules placed in the vicinity of silver clusters

We use a charge-dipole interaction model to study the polarizability of aromatic molecules that are placed between two silver clusters. In particular we examine the enhancement in polarizability induced by the clusters at plasmon-like resonant frequencies of the cluster-molecule-cluster system. The model used for these simulations relies on representation of the atoms by both a net electric charge and a dipole. By relating the time variation of the atomic charges to the currents that flow through the bonds of the structures considered, a least-action principle can be formulated that enables the atomic charges and dipoles to be determined. We consider benzene, naphthalene and anthracene for this study, comparing the polarizability of these aromatic molecules when placed in the middle between two Ag(120) clusters, with their polarizability as isolated molecules. We find that the polarizability of these molecules is enhanced by the clusters, and this increases the electromagnetic coupling between the two clusters. This results in significant red-shifting (by up to 0.8 eV) of the lowest energy optical transition in the cluster-molecule-cluster system compared to plasmon-like excitation in the cluster-cluster system. The resulting resonant polarizability enhancement leads to an electromagnetic enhancement in surface-enhanced Raman scattering of over 10(6).

Probing weakly bound molecules with nonresonant light

We show that weakly bound molecules can be probed by "shaking" in a pulsed nonresonant laser field. The field introduces a centrifugal term which expels the highest vibrational level from the potential that binds it. Our numerical simulations applied to the Rb(2) and KRb Feshbach molecules indicate that shaking by feasible laser pulses can be used to accurately recover the square of the vibrational wave function and, by inversion, also the long-range part of the molecular potential.

A charge-dipole interaction model for the frequency-dependent polarizability of silver clusters

We present a charge-dipole interaction model for the calculation of the frequency-dependent polarizability of silver clusters. The model relies on the representation of silver atoms by both a net electric charge and a dipole. Time variations of the atomic charges are related to the currents that flow through the bonds of the structures considered and the atomic charges and dipoles are eventually determined from the application of a least-action principle. After a generalization that enables the bonds of the bulk and surface atoms to have specific resistances, the model is parameterized on data obtained by the time-dependent density functional theory for tetrahedral Ag(20), Ag(84) and Ag(120) clusters. We then study the polarization properties of dimers of silver clusters. We compare in particular the polarizability of the dimers with that of the isolated clusters, for a range of gap distances and frequencies. We also consider the field enhancements one can achieve with these systems. The results are in good agreement with reference data and enable an extension of these data to a wider range of situations. They show that significant field enhancements are achieved at frequencies associated with resonant polarization along the axis of the dimer.

Accurate Molecular Polarizabilities Based on Continuum Electrostatics

A novel approach for representing the intramolecular polarizability as a continuum dielectric is introduced to account for molecular electronic polarization. It is shown, using a finite-difference solution to the Poisson equation, that the Electronic Polarization from Internal Continuum (EPIC) model yields accurate gas-phase molecular polarizability tensors for a test set of 98 challenging molecules composed of heteroaromatics, alkanes and diatomics. The electronic polarization originates from a high intramolecular dielectric that produces polarizabilities consistent with B3LYP/aug-cc-pVTZ and experimental values when surrounded by vacuum dielectric. In contrast to other approaches to model electronic polarization, this simple model avoids the polarizability catastrophe and accurately calculates molecular anisotropy with the use of very few fitted parameters and without resorting to auxiliary sites or anisotropic atomic centers. On average, the unsigned error in the average polarizability and anisotropy compared to B3LYP are 2% and 5%, respectively. The correlation between the polarizability components from B3LYP and this approach lead to a R2 of 0.990 and a slope of 0.999. Even the F2 anisotropy, shown to be a difficult case for existing polarizability models, can be reproduced within 2% error. In addition to providing new parameters for a rapid method directly applicable to the calculation of polarizabilities, this work extends the widely used Poisson equation to areas where accurate molecular polarizabilities matter.

An electrostatic interaction model for frequency-dependent polarizability: methodology and applications to hydrocarbons and fullerenes

We present an electrostatic interaction model for the calculation of frequency-dependent electronic polarizability. The model is based on the representation of atoms by an induced electric charge and a dipole moment. The extension of the charge-dipole model to the interaction with oscillating electric fields requires us to account for the kinetics of the free moving charges and of the atomic dipoles. This step is achieved by first relating the oscillations of the atomic charges to the currents that flow through the atomic bonds. Adopting a classical description of this system of charges and dipoles, the time evolution of these quantities is determined from the principle that the action be minimized. As an application of this model, we compute the frequency-dependent molecular polarizability of hydrocarbons with different hybridization. We propose parameters for each chemical group, which provide an excellent agreement with reference data provided by current-density-functional theory. We finally study the frequency-dependent molecular polarizability of fullerenes ranging from C(60) to C(1500).