Computer based methods implicit method

The theoretical tools of quantum chemistry briefly described in the previous chapter are numerously implemented, sometimes explicitly and sometimes implicitly, in ab initio, density functional (DFT), and semi-empirical theories of quantum chemistry and in the computer program suits based upon them. It is usually believed that the difference between the methods stems from different approximations used for the one- and two-electron matrix elements of the molecular Hamiltonian eq. (1.177) employed throughout the calculation. However, this type of classification is not particularly suitable in the context of hybrid methods where attention must be drawn to the way of separating the entire molecular system (eventually - the universe itself) into parts, of which some are treated explicitly on a quantum mechanical/chemical level, while others are considered classically and the rest is not addressed at all. That general formulation allows us to cover both the traditional quantum chemistry methods based on the wave functions and the DFT-based methods, which generally claim... 

All computations are performed using the code OpenFOAM [17], an open source computational fluid dynamics (CFD) toolbox, utilizing a cell-center-based finite volume method on a fixed unstructured numerical grid and employing the solution procedure based on the pressure implicit with splitting of operators (PISO) algorithm for coupling between pressure and velocity in transient flows [18]. 

In section 9.2 we illustrated one explicit method, Euler s forward method. In the present section, we likewise used only one type of implicit method, based on the trapezoidal or midpoint rule. All our examples have used constant increments Af higher computational efficiency can oftenbe obtained by making the step size dependent on the magnitudes of the changes in the dependent variables. Still, these examples illustrate that, upon comparing equivalent implicit and explicit methods, the former usually allow larger step sizes for a given accuracy, or yield more accurate results for the same step size. On the other hand, implicit methods typically require considerably more initial effort to implement. 

In order to set up an MD simulation based on the implicit solvation model, a set of atomic radii is required, which is an additional set of input parameters compared to the explicit solvent case. A number of implicit solvent models have been developed, and the effect of the solvent is treated as an average potential acting on the solute. The implicit solvent model based on a finite difference solution of the Poisson-Boltzmann (PB) theory provides a rigorous theoretical framework and captures the polar component of the free energy of solvation for a given biomolecule quite well.35 However, it is a computationally expensive method and, therefore, has limited application in MD simulations. 

Raw computational speed has been considered one of the key advantages of the GB model. However, note that the cost of a calculation based directly on Eq. (3) is generally O(N ) for a system of N atoms, while the scaling is more favorable, N log(N), for Ewald-based methods used in explicit solvent simulations. For large systems, e.g. the nucleosome (25,000 atoms), the number of nanoseconds of MD per GPU hour may actually be less in a GB-based simulation (without additional approximations such as cut-offs) than in a comparable explicit solvent run [54], although the conformational search is still much faster in the implicit solvent. 

The triangle relation of Equation 13.6 is a strong and most useful constraint for computation. The methods for calculation of pK and redox potentials must be consistent and satisfy Equation 13.6 by construction. Quantum chemistry schemes based on implicit solvent models normally meet this consistency requirement. " For all-atom schemes, this is much harder to achieve. This is ultimately why we decided to compute pK by completely removing the acid proton from solution - rather than transferring it to the solvent, which would be the natural approach in DFTMD.35.36... 

It should be kept in mind that an objective function which does not require any phase equilibrium calculations during each minimization step is the basis for a robust and efficient estimation method. The development of implicit objective functions is based on the phase equilibrium criteria (Englezos et al. 1990a). Finally, it should be noted that one important underlying assumption in applying ML estimation is that the model is capable of representing the data without any systematic deviation. Cubic equations of state compute equilibrium properties of fluid mixtures with a variable degree of success and hence the ML method should be used with caution. 

The computational code used in solving the hydrodynamic equation is developed based on the CFDLIB, a finite-volume hydro-code using a common data structure and a common numerical method (Kashiwa et al., 1994). An explicit time-marching, cell-centered Implicit Continuous-fluid Eulerian (ICE) numerical technique is employed to solve the governing equations (Amsden and Harlow, 1968). The computation cycle is split to two distinct phases a Lagrangian phase and a remapping phase, in which the Arbitrary Lagrangian Eulerian (ALE) technique is applied to support the arbitrary mesh motion with fluid flow. 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

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