Extended Lagrangian method

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

A different predictive procedure is to use the extended Lagrangian method, in which each dipole is treated as a dynamical variable and given a mass M and velocity (i. The dipoles thus have a kinetic energy, and are propagated using the equations of motion just like the atomic coordi-nates. The equation of motion for the dipoles is... 

In most electronegativity equalization models, if the energy is quadratic in the charges (as in Eq. [36]), the minimization condition (Eq. [41]) leads to a coupled set of linear equations for the charges. As with the polarizable point dipole and shell models, solving for the charges can be done by matrix inversion, iteration, or extended Lagrangian methods. 

In the extended Lagrangian method, as applied to a fluctuating charge system,the charges are given a fictitious mass, M, and evolved in time according to Newton s equation of motion, analogous to Eq. [23],... 

Regardless of the type of model used, a method must be chosen for the self-consistent solution of the polarizable degrees of freedom. Direct solution via matrix inversion is nearly always avoided by most researchers in the field, because of the prohibitive O(N ) scaling with system size, N. Both iterative and predictive methods reduce the scaling to match that of the potential evaluation [O(N ) for direct summation 0(N In N) for Ewald-based meth-ods ° " 0 N) if interactions are neglected beyond some distance cutoff], but the cost of the iterations means that the predictive methods are always more efficient. Extended Lagrangian methods have been implemented for all four types of polarizable... 

Dynamics Simulation of Polarizable Water by an Extended Lagrangian Method. 

The extended Lagrangian method is the base for the development of two modified version of SPC and TIP4P, named SPCfq and TIP4Pfq, where the charge itself instead of the induced dipole, is treated as a dynamical variable [100], while the molecular geometry and the number of sites is the same as in the parent functions. 

The essence of the extended Lagrangian method [73, 74] is to choose small enough fictitious masses m in the multidimensional version of Eq. (3.79) that the fictitious variables rapidly oscillate around the minimum of the centroid free-energy surface [Eq. (3.78)]. The centroid variables should then exhibit an adiabatic, conservative motion because there is little energy exchange between them and the fictitious degrees of... 

In the following the indices for the nuclei and the electrons will be omitted. Applying the extended Lagrangian method introduced by Niklasson [7, 8] a general expression for the AIMD Lagrangian can be written as... 

D. van Belle, M. Froeyenand, G. Lippens, and S. J. Wodak, Mol. Phys., 77, 239 (1992). Molecular Dynamics Simulation of Polarizable Water by an Extended Lagrangian Method. 

In a spirit similar to the extended Lagrangian method, an additional degree of freedom can be added to the position of all particles in the system. The position of an atom becomes (x, y, z, w) and the Euclidean metric distance between two particles is taken in a four-dimensional space... 

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