Car-Parrinello MD method

A group of theoretical methods exists where the electronic wavefuntion is computed, and the atomic nuclei are propagated (using classical equations of motion). The Car-Parrinello MD method is one of this type [22-24]. These methods he between the extremes of the classical and ab initio methods, as they include some (quantum) electronic information and some (classical) dynamics information. These methods are called ah initio or first principles MD if you come from the classical community and semi-classical MD if you come firom the quantum community [9], Ah initio MD methods are far more expensive and cannot simulate as many molecules for as long as the classical simulations, but they are more flexible in that structures are not predetermined and information on the electronic structure is retained. Semi-classical MD can be carried out under periodic boundary conditions and thus the local liquid environment, and any extended bonding network, vyill be present. These methods hold a great deal of promise for the future study of ionic liquid systems, the first such calciilations on ionic liquids were reported in 2005 [21,25]. 

On the other hand, the FE of activation of the inverse reaction, AAjj, was 1.8 kcal moF (MP2) and is in good agreement with the estimation by the Car-Parrinello MD method ( 1.5 kcal moF [40]). On the contrary, the FE overestimation at the HF level could be understood mainly due to the incorrectness of the PE itself as shown in Fig. 8.8. It is understood that, for discussion of the FE stability of the proton transfer reaction of glycine in aqueous solution, it is essentially necessary to include the electron correlation effect at least by the MO calculation at the MP2 level. 

Exner and coworkers have used Af-methyl acetamide (NMA) as a test system for amide groups in protein backbones to calculate NMR chemical shifts with the Car-Parrinello MD method with explicit solvent molecules and quantum-chemical calculations of NMR parameters and compare with classical MD simulations. For example the C-P calculations give in general shorter solute-solvent H-bonds which in turn give a... 

Reaction rates are macroscopic averages of the number of microscopical molecules that pass from the reactant to the product valley in the potential hypersurface. An estimation of this rate can be obtained from the energy of the highest point in the reaction path, the transition state. This approach will however fail when the reaction proceeds without an enthalpic barrier or when there are many low frequency modes. The study of these cases will require the analysis of the trajectory of the molecule on the potential hypersurface. This idea constitutes the basis of molecular dynamics (MD) [96]. Molecular dynamics were traditionally too computationally demanding for transition metal complexes, but things seem now to be changing with the use of the Car-Parrinello (CP) method [97]. This approach has in fact been already succesfully applied to the study of the catalyzed polymerization of olefins [98]. 

With the development of GGA functionals, description of molecular systems with the Kohn-Sham method reached a precision similar to other quantum theory methods. It was quickly shown that the GGA s could also well reproduce the hydrogen bond properties. Short after, liquid water at ambient condition was first simulated by Car-Parrinello MD, with a sample of 32 water molecules with periodic boundary conditions [31]. Since then, many simulations of liquid water at different temperatures and pressures and of water solutions have been performed [32-39]. Nowadays, Car-Parrinello MD has become a major tool for the study of aqueous solutions [40-64]. 

Frank and coworkers have employed a constrained distance method in conjunction with Car-Parrinello MD (CPMD) simuIatiOTis [99] to examine a variety of mechanochemical processes [33, 39-41, 100, 101]. During these simulations, F is applied by increasing the distance between a pair of atoms at a cmistant velocity. They have used this approach to explore the changes in electronic structure that occur when solvated polymers are stretched to the point of rupture [41]. Their studies showed that bond rupture occurred through a heterolytic process involving solvent molecules. Interestingly, their simuIatiOTis showed that the weakest bond does not necessarily correspond to the site of bond rupture. Rather, the bmid that is made most accessible to attack by solvent via E-induced changes in structure most ft equently corresponds to the site at which the polymer dissociates. 

Classical force field MD, MC Embedded atomistic-quantum methods Car-Parrinello MD Quantum MC... 

Hybrid methods learn from the pros and cons of classical and quantum methods and try to combine the best features of both. These methods are of two categories (1) methods that are hybrid in a temporal sense, mixing electronic structure calculations with MD methods and (2) methods that are spatially hybrid, applying different methods in different physical regions of a molecule or computational domain. Car-Parrinello MD (CPMD) (Car and Parrinello 1985) and related methods such as... 

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].)... 

In 1985 Car and Parrinello invented a method [111-113] in which molecular dynamics (MD) methods are combined with first-principles computations such that the interatomic forces due to the electronic degrees of freedom are computed by density functional theory [114-116] and the statistical properties by the MD method. This method and related ab initio simulations have been successfully applied to carbon [117], silicon [118-120], copper [121], surface reconstruction [122-128], atomic clusters [129-133], molecular crystals [134], the epitaxial growth of metals [135-140], and many other systems for a review see Ref. 113. 

An alternative approach was introduced by Car and Parrinello [12], who developed a DFT-MD method to study periodic systems using a planewave expansion in which the electronic parameters, as well as the nuclear coordinates, are treated as dynamical variables. Following the Car and Parrinello... 

At several points in diis book, it has been emphasized that the prevalence of classical MC and MD simulations derives from die impracticality of carrying out fully QM dynamics. While diis is largely true, for systems of only modest size where short trajectories may be profitably analyzed, fully QM MD simulations using the so-called Car-Parrinello technique are a viable option (Car and Parrinello 1985). In its most widely used formulation, the Car-Parrinello method employs DFT as the electronic-structure method of choice. In... 

Hybrid multiscale models enable us to focus on the relevant part of a system. For example, Leenders et al. studied the proton transfer process in the photoactive yellow protein (Figure 6.3) [9], They used Car-Parrinello molecular dynamics [10], a QM method for dynamics simulations, to describe the chromophore and its hydrogen-bonded network in the protein pocket (middle and right-hand circles). This was combined with a traditional MD force field of 28 600 atoms, simulating the entire protein in water (left-hand circle). 

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]. 

The Intention of this volume is to give a flavour of the types of problems in biochemistry that theoretical calculations can solve at present, and to illustrate the tremendous predictive power these approaches possess. With these aspects in mind, I have tried to gather some of the leading scientists in the field of theoretical/computational biochemistry and let them present their work. You will hence find a wide range of computational approaches, from classical MD and Monte Carlo methods, via semi-empirical and DFT approaches on isolated model systems, to Car-Parrinello QM-MD and novel hybrid QM/MM studies. The systems investigated also cover a broad range from membrane-bound proteins to various types of enzymatic reactions as well as inhibitor studies, cofactor properties, solvent effects, transcription and radiation damage to DNA. 

Similiar problems are known in classical MD simulations, where intramolecular and intermolecular dynamics evolve on different time scales. One possible solution to this problem is the method of multiple time scale propagators which is describede in section 5. Berne and co-workers [21] first used different time steps to integrate the intra- and intermolecular degrees of freedom in order to reduce the computational effort drastically. The method is based on a Trotter-factorization of the classical Liouville-operator for the time evolution of the classical system, resulting in a time reversible propagation scheme. The multiple time scale approach has also been used to speed up Car-Parrinello simulations [20] and ab initio molecular dynamics algorithms [21]. 

In summary, this method solves the Schrodinger equation at several intervals of time for the largest possible sample that can be solved with present computational resources. It also creates a force field to compute forces with a classical molecular dynamics procedure in a system containing the largest number of particles that is practical to be used with MD methods. When the time intervals of the ab initio calculations coincide with the time intervals of the molecular dynamics, and when the electron density distribution is used to compute the forces instead of the force field, this method is equivalent to the well known Car-Parrinello method. Evidently, this latter method is limited to a... 

The Car-Parrinello (CP) approach is an extended energy minimization and MD scheme in which electronic degrees of freedom are treated explicitly on an equal footing with nuclear coordinates. Below we outline the conceptual scheme of the method and refer the interested reader elsewhere for a comprehensive discussion. " ... 

The purpose of this chapter will be to review the fundamentals of ab initio MD. We will consider here Density Functional Theory based ab initio MD, in particular in its Car-Parrinello version. We will start by introducing the basics of Density Functional Theory and the Kohn-Sham method, as the method chosen to perform electronic structure calculation. This will be followed by a rapid discussion on plane wave basis sets to solve the Kohn-Sham equations, including pseudopotentials for the core electrons. Then we will discuss the critical point of ab initio MD, i.e. coupling the electronic structure calculation to the ionic dynamics, using either the Born-Oppenheimer or the Car-Parrinello schemes. Finally, we will extend this presentation to the calculation of some electronic properties, in particular polarization through the modern theory of polarization in periodic systems. 

Section 6 will deal with the core of the problem at hand, coupling of the electronic calculation with Molecular Dynamics for ions. We will describe in particular the Car-Parrinello method introduced in 1985 [2], which is really the starting point for combining on the fly electronic structure calculation to MD and statistical physics. This will be illustrated rapidly by a simulation of 32 water molecules at room temperature. 

Having presented the Car-Parrinello method and discussed some of its properties, we can now try to assess its advantages and disadvantages with respect to the Born-Oppenheimer MD. 

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