Car-Parrinello lagrangian

Car and Parrinello in their celebrated 1985 paper [2] proposed an alternative route for molecular simulations of electrons and nuclei altogether, in the framework of density functional theory. Their idea was to reintroduce the expansion coefficients Cj(G) of the Kohn-Sham orbitals in the plane wave basis set, with respect to which the Kohn-Sham energy functional should be minimized, as degrees of freedom of the system. They then proposed an extended Car-Parrinello Lagrangian for the system, which has dependance on the fictitious degrees of freedom Cj(G) and their time derivative Cj (G) ... 

The Car-Parrinello method presents a trick to propagate the orbitals along with the nuclei without reoptimizing them at each molecular dynamics step. It does so by introducing a fictitious orbital mass and propagating the orbitals with the nuclei via appropriate equations of motion. The Car-Parrinello Lagrangian is... 

Prove the expressions for the various terms in the energy of a Frenkel exciton represented by a Slater determinant of Wannier functions, given in Eq. (5.65). Derive the equations of motion for the electronic and ionic degrees of freedom from the Car-Parrinello lagrangian, Eq. (5.98). 

Although constrained dynamics is usually discussed in the context of the geometrically constrained system described above, the same techniques can have many other applications. For instance, constant-pressure and constant-temperature dynamics can be imposed by using constraint methods [33,34]. Car and Parrinello [35] describe the use of the extended Lagrangian to maintain constraints in the context of their ab initio MD method. (For more details on the Car-Parrinello method, refer to the excellent review by Gain and Pasquarrello [36].)... 

The Kohn-Sham theory made a dramatic impact in the field of ab initio molecular dynamics. In the 1985, Car and Parrinello38 introduced a new formalism to study dynamics of molecular systems in which the total energy functional defined as in the Kohn-Sham formalism proved to be instrumental for practical applications. In the Car-Parrinello method (CP), the equations of motion are based on a Lagrangian (Lcp) which includes fictitious degrees of freedom associated with the electronic state. It is defined as ... 

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

Despite the simple form of Equation (1.83), the detailed formulation of an extended Lagrangian for CPCM is not a straightforward matter and its implementation remains challenging from the technical point of view. Nevertheless, is has been attempted with some success by Senn and co-workers [31] for the COSMO-ASC model in the framework of the Car-Parrinello ab initio MD method. They were able to ensure the continuity of the cavity discretization with respect to the atomic positions, but they stopped short of providing a truly continuous description of the polarization surface charge as suggested,... 

In order to overcome the limitations of currently available empirical force field param-eterizations, we performed Car-Parrinello (CP) Molecular Dynamic simulations [36]. In the framework of DFT, the Car-Parrinello method is well recognized as a powerful tool to investigate the dynamical behaviour of chemical systems. This method is based on an extended Lagrangian MD scheme, where the potential energy surface is evaluated at the DFT level and both the electronic and nuclear degrees of freedom are propagated as dynamical variables. Moreover, the implementation of such MD scheme with localized basis sets for expanding the electronic wavefunctions has provided the chance to perform effective and reliable simulations of liquid systems with more accurate hybrid density functionals and nonperiodic boundary conditions [37]. Here we present the results of the CPMD/QM/PCM approach for the three nitroxide derivatives sketched above details on computational parameters can be found in specific papers [13]. 

Formally, the fictitious dynamics of Car-Parrinello (CP) can be derived from the following Lagrangian [4—6]... 

The extended Lagrangian technique on which the Car-Parrinello method is based can be used also in other contexts. Whenever the forces on some atoms... 

First-principles simulations are techniques that generally employ electronic structure calculations on the fly . Since this is a very expensive task in terms of computer time, the electronic structure method is mostly chosen to be density functional theory. Apart from the possibility of propagating classical atomic nuclei on the Born-Oppenheimer potential energy surface represented by the electronic energy V (R ) = ji(R ), another technique, the Car-Parrinello method, emerged that uses a special trick, namely the extended Lagrangian technique. The basic idea... 

The essential trick of the Car-Parrinello method is not to parametrize the potential-energy surface, but to calculate it on the fly from first principles when generating the nuclear trajectories by Newton s mechanics. The electronic groimd state is calculated at the same time. To do so, one starts with an extended Lagrangian which corresponds to a fictitious dynamical system... 

The hybrid QM/MM Car-Parrinello method is based on a mixed Lagrangian of the form... 

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... 

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