Computational advantages

This Legendre expansion converges rapidly only for weakly anisotropic potentials. Nonetheless, truncated expansions of this sort are used more often than justified because of their computational advantages. 

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

MCSCF theory is a specialist branch of quantum modelling. Over the years Jt has become apparent that there are computational advantages in treating all oossible excitations arising by promoting electron(s) from a (sub)set of the occu-orbitals to a (sub)set of the virtual orbitals. We then speak of complete active ace MCSCF, or CASSCF. 

The CPHF equations are linear and can be determined by standard matrix operations. The size of the U matrix is the number of occupied orbitals times the number of virtual orbitals, which in general is quite large, and the CPHF equations are normally solved by iterative methods. Furthermore, as illustrated above, the CPHF equations may be formulated either in an atomic orbital or molecular orbital basis. Although the latter has computational advantages in certain cases, the former is more suitable for use in connection with direct methods (where the atomic integrals are calculated as required), as discussed in Section 3.8.5. 

CS INDO [10] (as well as the parent C INDO [9]) shares the same basic idea as the PCILO scheme [29,30] to exploit the conceptual and computational advantages of using a basis set of hybrid atomic orbitals (AOs) directed along, or nearly, the chemical bonds. 

The hierarchical methods so far discussed are called agglomerative. Good results can also be obtained with hierarchical divisive methods, i.e., methods that first divide the set of all objects in two so that two clusters result. Then each cluster is again divided in two, etc., until all objects are separated. These methods also lead to a hierarchy. They present certain computational advantages [21,22]. 

For one specific set of discharge parameters, in a comparison between the hybrid approach and a full PIC/MC method, the spectra and the ion densities of the hybrid model showed some deviations from those of the full particle simulation. Nevertheless, due to its computational advantages, the hybrid model is appropri-... 

In our opinion the above formulation does not provide any computational advantage over the approach described next since the PDEs (state and sensitivity equations) need to be solved numerically. 

To further illustrate the conceptual and computational advantages offered by the moving dividing surface, extensive simulations of several quantities relevant to rate theory calculations were performed [39] on the anharmonic model potential... 

The coupled DFT/MM formalism can be regarded as an intermediate approximation between ab initio molecular dynamics, and classical molecular mechanics. Being so, the range of its applicability extends to problems not treatable by molecular mechanics, chemical reactions for instance. The possibility of restricting quantum-mehcanical treatment to well-localized regions also makes it computationally advantageous over supermolecule ab initio simulations. It is important to note that this formalism does not differ whether applied to study biochemical reactions or to study reactions taking place in an other microscopic environment. This makes it possible to test any implementation on problems for which there... 

For a quantitative description of molecular geometries (i.e. the fixing of the relative positions of the atomic nuclei) one usually has the choice between two possibilities Cartesian or internal coordinates. Within a force field, the potential energy depends on the internal coordinates in a relatively simple manner, whereas the relationship with the Cartesian nuclear coordinates is more complicated. However, in the calculations described here, Cartesian coordinates are always used, since they offer a number of computational advantages which will be commented on later (Sections 2.3. and 3.). In the following we only wish to say a few words about torsion angles, since it is these parameters that are most important for conformational analysis, a topic often forming the core of force field calculations. 

To conclude this sub-section, we note that the Cl form of the wave function (2.2) leads, via the mapping (2.6) from the coefficients c, to the charge distribution weights wm, to a computationally advantageous matrix formulation of the free energy of the solute plus solvent system, which we present next. 

In spite of many computational advantages, DPD and SCMF methods are not able accurately to predict physical properties that rely upon time correlation functions (e.g., diffusion), making them less applicable to extract structure-related transport properties of phase-segregated membranes. 

Equation (15.23) displays the feature of locality that the blending functions should possess in order to be computationally advantageous that is, during the process of matrix inversion, one wishes the calculation to proceed quickly. As mentioned earlier, the use of linear approximation functions results in at most five terms on the left side of the equation analogous to (15.23), yielding a much crader approximation, but one more easily calculated. The current choice of Bezier functions, on the other hand, is rapidly convergent for methods such as relaxation, possesses excellent continuity properties (the solution is guaranteed to look and behave reasonably), and does not require substantial computation. 

There are computational advantages to be realized by transforming the cross-stream coordinate to the stream function, namely the Von Mises Transformation. In a subsequent section this transformation is done for the general boundary-layer setting, and it includes... 

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