Classical Drude oscillator model

In the absence of an electric field, the Drude particle coincides with the position of the atom, r, and the atom appears as a point charge of magnitude q. In the presence of a uniform electric field E, the Drude particle assumes a displaced position r + d. The Drude separation d is related to kp, E and qp  

From which results a simple expression for the isotropic atomic polarizability  

The dependence on the nuclear positions is indicated by r and the dependence on the Drude positions is indicated by d. In Eq. (9-25) Ubond (r) is the intramolecular energy contribution from, typically, the bond lengths, valence angles, and dihedral angles, Ulj (r) is a Lennard-Jones 6-12 nonpolar contribution, Ueiect (r, d) represents all Coulombic interactions, atom-atom, atom-Drude, and Drude-Drude, and Useif (d) represents the atom-Drude harmonic bonds. The term Usey (d) arises from the harmonic spring separating the two charges and has the simple expression 

The electrostatic interaction between independent polarizable atoms is simply the sum of the charge-charge interactions between the four charge sites (i.e. two atoms and their respective Drude particles)  

For a given a the force constant ko can be chosen in a way that the displacement d of the Drude particle remains much smaller than the interatomic distance. This guarantees that the resulting induced dipole jl, is almost equivalent to a point dipole. In the Drude polarizable model the only relevant parameter is the combination q /ko which defines the atomic polarizability, a. It is 

The model was originally proposed by Paul Drude in 1902 as a simple way to describe dispersive properties of materials [108]. A quantum version of the model (including the zero-point vibrations of the oscillator) has been used in early applications to describe the dipole-dipole dispersion interactions [109-112]. A semiclassical version of the model was used more recently to describe molecular interactions [113], and electron binding [114]. The classical version has been subsequently used for ionic crystals [115-120], simple liquids [121-127], water [128-135], and ions [136-139], and in recent decades has seen widespread use in MD and MC simulations. In recent years, the Drude model was extended to interface with QM approaches in QM/MM methods [140], facilitated by the simplicity of the model in that it only includes additional charge centers. 

the expression for the induced atomic dipole, fi, as a function of d is 

In the Drude polarizable model, the only relevant adjustable parameter is the combination q /KD that corresponds to the atomic polarizability. In the limit of large Kd, the treatment of induced polarization based on Drude oscillators is formally equivalent to a point-dipole treatment such as used by AMOEBA. In practice, the magnitude of Kd is commonly chosen to achieve small displacements of Drude particles from their corresponding atomic positions, as required to remain close to the point-dipole approximation for the induced dipole associated with the atom-Drude pair [150] while preserving a stable integration of the equation of motion with a reasonable time step. For a fixed force constant Kd the atomic polarizability is determined by the amount of chaise assigned to the Drude particle. In the current implementation, the classical Drude model introduces atomic polarizabilities only to non-hydrogen atoms for practical considerations, as discussed below. However, this is adequate to accurately reproduce molecular polarizabilties, as seen in a number of published studies [127,142,146]. 

The total potential energy of the Drude polarizable model contains the terms representative of the interaction with the static electric field, interaction with other dipoles and the self-energy associated with the Drude oscillators, in addition to the standard contributions representing bonding terms (bonds, angles, dihedrals, etc.) and intermolecular interactions represented by Lennard-Jones (Lj) 6-12 term  

In this expression, the dependences on the nuclear positions and the displacements of the Drude particle are indicated by r and d, respectively, f/bond (r) and f/Lj(r) are, respectively, intramolecular bonding and non-bonding LJ energetic contributions. l/eiec(r d) is the sum over all Coulombic interactions between atomic core charges located at r[j and the Drude charges —and located at rjj and r ) respectively. The displacement vector for the Drude particle 

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Vorobyov IV, VM Anisimov, AD MacKerell Jr (2005) Polarizable empirical force field for alkanes based on the classical drude oscillator model. J. Phys. Chem. B 109 (40) 18988-18999... 

Explicit Inclusion of Induced Polarization in Atomistic Force Fields Based on the Classical Drude Oscillator Model... 

Savelyev, A., and A. D. MacKerell, All-atom polarizable force field for DNA based on the classical drude oscillator model/. Comput. Chem., 2014. 35(16), 1219-1239. 

Lemkul, J. A., Roux, B., van der Spoel, D and MacKerell, A. D., Jr., Implementation of Extended Lagrangian Dynamics in GROMACS for Polarizable Simulations Using the Classical Drude Oscillator Model, In Press./ Comput. Chem., 2015. 36,1480-1486. 

The total electric field, E, is composed of the external electric field from the permanent charges E° and the contribution from other induced dipoles. This is the basis of most polarizable force fields currently being developed for biomolecular simulations. In the present chapter an overview of the formalisms most commonly used for MM force fields will be presented. It should be emphasized that this chapter is not meant to provide a broad overview of the field but rather focuses on the formalisms of the induced dipole, classical Drude oscillator and fluctuating charge models and their development in the context of providing a practical polarization model for molecular simulations of biological macromolecules [12-21], While references to works in which the different methods have been developed and applied are included throughout the text, the major discussion of the implementation of these models focuses... 

From this point of view it is of interest to examine the consequences of full ther-malization of the classical Drude oscillators on the properties of the system. This is particularly important given the fact that any classical fluctuations of the Drude oscillators are a priori unphysical according to the Bom-Oppenheimer approximation upon which electronic induction models are based. It has been shown [12] that under the influence of thermalized (hot) fluctuating Drude oscillators the corrected effective energy of the system, truncated to two-body interactions is... 

Lamoureux G, MacKerell AD, Roux B (2003) A simple polarizable model of water based on classical Drude oscillators. J Chem Phys 119(10) 5185—5197... 

Lamoureux G, Roux B (2003) Modeling induced polarization with classical Drude oscillators theory and molecular dynamics simulation algorithm. J Chem Phys 119(6) 3025-3039... 

Harder E, Anisimov VM, Vorobyov IV, Lopes PEM, Noskov SY, MacKerell AD, Roux B (2006) Atomic level anisotropy in the electrostatic modeling of lone pairs for a polarizable force field based on the classical Drude oscillator. J Chem Theory Comput 2(6) 1587-1597... 

Current polarizable models can be classified into three major categories (1) induced dipole, [2] fluctuating charge, and [3] classical Drude oscillator (or Shell models]. 

Yu, W, et al.. Six-site polarizable model of water based on the classical Drude oscillator./ Chem. Phys., 2013.138(3], 034508. 

The induced dipole formulation is an example of the so-called polarizable embedding but it is not the only possible choice. There are in fact various alternatives schemes to simulate the polarization of the MM subsystem, such as the fluctuating charges [24, 25], the classical Drude oscillators [26], or the Electronic Response of the Surroundings (ESR) [27] which mixes a non-polarizable MM scheme with a polarizable continuum model characterized by a dielectric constant extrapolated at infinite frequency. 

MacKerell and coworkers presented a polarizable force field, called Drude-2013, based on the classical Drude oscillator framework, currently implemented in simulation programs CHARMM and NAMD, for modeling of peptides and proteins. It can be used for several hundreds of ns simulations without requiring too much computing power. In its first generation version it gives improved order parameters for certain residues, for which they were underestimated using additive force fields, but does not perform so well yet for NMR chemical shifts. 

Z. Y. Lu and Y. K. Zhang,/. Chem. Theory Comput., 4(8), 1237-1248 (2008). Interfacing Ab Initio Quantum Mechanical Method with Classical Drude Oscillator Polarizable Model for Molecular Dynamics Simulation of Chemical Reactions. 

The classical theory of absorption in dielectric materials is due to H. A. Lorentz and in metals it is the result of the work of P. K. L. Drude. Both models treat the optically active electrons in a material as classical oscillators. In the Lorentz model the electron is considered to be bound to the nucleus by a harmonic restoring force. In this manner, Lorentz s picture is that of the nonconductive dielectric. Drude considered the electrons to be free and set the restoring force in the Lorentz model equal to zero. Both models include a damping term in the electron s equation of motion which in more modem terms is recognized as a result of electron-phonon collisions. 

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