Extended system approach

In the preceding sections, we have presented simulation methodologies to study systems away from equilibrium. In particular, we have concentrated on the problem of shear flow in bulk systems. In the present section, we illustrate the effectiveness of the SLLOD dynamics coupled with the robustness of the extended system approach to initiate and sustain a shear flow in a fluid in the absence of moving boundary conditions. 

In the extended system approach, additional degrees of freedom that control the quantity under consideration can be added to the system [95, 96, 103, 104], Thereby thermostats or barostats can be simulated [105-107], which provide a quality of modeling thermodynamic processes hardly reached by the stationary quantum chemical methods discussed in the previous two sections. 

Two methods that have been developed that do maintain correct canonical averaging are the stochastic collision and the extended systems approaches. Both are covered in detail elsewherel . We report here only on some of the salient features from the extended systems approach since this approach is used primarily for constant temperature MD simulations for heterogeneous catalytic materials. 

Applying Flartree-Fock wavefiinctions to condensed matter systems is not routine. The resulting Flartree-Fock equations are usually too complex to be solved for extended systems. It has been argried drat many-body wavefunction approaches to the condensed matter or large molecular systems do not represent a reasonable approach to the electronic structure problem of extended systems. 

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 algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

A variety of l,2,3,4-tetrahydro-j8-carbolines have been prepared from 3-piperidone phenylhydrazone derivatives. Used initially to obtain pentacyclic derivatives (35) related to the yohimbe alkaloids, this route was later extended to the synthesis of tetracyclic compounds (36). l-Methyl-5,6,7,8-tetrahydro-j8-carboline (37) was prepared in low yield by heating cyclohexanone 2-methyl-3-pyridylhydrazone with zinc chloride, a synthesis probably based on the similar preparation of the tetracyclic compound 38 starting from the corresponding quinolylhydrazine. Abramovitch and Adams extended this approach to the cyclization of cyclohexanone 3-pyri-dylhydrazone (39) itself. The main product was 6,7,8,9-tetrahydro-8-carboline (40), a smaller amount of the j8-isomer (41) also being obtained. This provides a convenient and readily reproducible route to the otherwise difficultly accessible 8-carboline ring system. The favored attack at carbon-2 over carbon-4 of the pyridine nucleus... 

In comparison with the more standard Configuration Interaction (Cl) method, the one-particle Green s function approach offers the essential advantages, in the outlook of numerical applications on extended systems, of a stronger and systematic compactness (30) of the configuration spaces in high order approximations and of energy separability (5,31) in the dissociation limit (size-consistency). The latter is a necessary prerequisite ( ) for a correct (i.e. size-... 

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... 

Dithiolene complexes with the maleonitriledithiolate (mnt) ligand form highly delocalized systems and are widespread in studies of conducting and magnetic materials. The electronic properties have been extensively studied with various computational methods including Hiickel and extended Hiickel approaches to identify the nature of the orbitals involved in intramolecular and intermolecular interactions. These structural properties allow the complexes to interact in the solid state via short stacking S, S and short interstack S---S contacts.10 4-1048... 

During the last decade, density-functional theory (DFT)-based approaches [1, 2] have advanced to prominent first-principles quantum chemical methods. As computationally affordable tools apt to treat fairly extended systems at the correlated level, they are also of special interest for applications in medicinal chemistry (as demonstrated in the chapters by Rovira, Raber et al. and Cavalli et al. in this book). Several excellent text books [3-5] and reviews [6] are available as introduction to the basic theory and to the various flavors of its practical realization (in terms of different approximations for the exchange-correlation functional). The actual performance of these different approximations for diverse chemical [7] and biological systems [8] has been evaluated in a number of contributions. 

Hughmark (Hll) has extended this approach to obtain an empirical correlation covering wide ranges of data for the air-water systems in vertical flow. Basically the correlation consists of using Eq. (70) with a variable value of the coefficient K. This coefficient was expressed by Hughmark as a function of the mixture Reynolds number, Froude number, and liquid volume-fraction. Hughmark s approach gives... 

Explicitly correlated wave functions described above have been specifically designed for two-electron molecular systems. As it was demonstrated in the previous section these functions give the energies which appear to be superior to the variational energies reported. Therefore several attempts have been made to extend this approach to many-electron molecules. The James-Coolidge (JC) type of function has been extended to three- and four-electron diatomic molecules by Clary and... 

To estimate the consequences of such a bridge modification in a quantitative fashion one needs to treat an extended system where donor and acceptor sites are separated by intervening WCPs. We used the effective Hamiltonian approach described in Sect. 2.7 to clarify the distance dependence of the electronic coupling in DNA and the effect of bridge modifications [50]. 

One interesting scheme based on density functional theory (DFT) is particularly appealing, because with the current power of the available computational facilities it enables the study of reasonably extended systems. DFT has been applied with a variety of basis sets (atomic orbitals or plane-waves) and potential formulations (all-electron or pseudopotentials) to complex nu-cleobase assemblies, including model systems [90-92] and realistic structures [58, 93-95]. DFT [96-98] is in principle an ab initio approach, as well as MP2//HF. However, its implementation in manageable software requires some approximations. The most drastic of all the approximations concerns the exchange-correlation (xc) contribution to the total DFT functional. 

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