Grid-based methods

Besides the aforementioned descriptors, grid-based methods are frequently used in the field of QSAR quantitative structure-activity relationships) [50]. A molecule is placed in a box and for an orthogonal grid of points the interaction energy values between this molecule and another small molecule, such as water, are calculated. The grid map thus obtained characterizes the molecular shape, charge distribution, and hydrophobicity. 

Thirdly, the binary-tree tabulation algorithm used in ISAT is very different from the grid-based method described above for pre-computed lookup tables. We will look at each of these aspects in detail below. However, we will begin by briefly reviewing a few points from non-linear systems theory that will be needed to understand ISAT. 

As in the case of photodissociation, the quantum theory of reactive molecular scattering was initially entirely based on time-independent scattering theory [4-7,100-123]. There were several early attempts to apply time-dependent quantum theory to reactive scattering processes [124—131]. But the modern era of the field really began with the seminal work of Kosloff et al. [37] and the subsequent application of his grid-based methods to the reactive scattering problem by Neuhauser and Baer and coworkers [45,132]. There have been many developments in the field [93,133-138], and several reviews and a book have been written on the topic [10,139-141]. My aim in the next section will be to outline the basic methods of time-dependent quantum theory used in reactive scattering calculations. While the review will cover many aspects of the theory, it will not cover all the approaches currently in use (as of 2003). 

Substituting Eq. (21) into the MEP function analogue of Eq. (15), the MEP-SI can be calculated analytically without application of an arbitrary grid. This method was faster by two orders of magnitude than the corresponding grid-based method and the calculated MEP-Si(C) values were in very good correlation with each other [113]. 

Unlike the classical —> autocorrelation descriptors, only the highest product of interaction energies per distance bin is stored as GRIND descriptor (MACC-2 transform). This difference is responsible for the reversibility of GRIND descriptors. Unlike most of the grid-based methods, GRIND descriptors are also independent of the molecule alignment. 

Clustering algorithms can be classified into four major approaches hierarchical methods, partitioning-based methods, density-based methods, and grid-based methods. Here, we will focus on the hierarchical cluster approach because it is often used in the context of structure-activity analysis. Recent research has suggested that hierarchical methods perform better than the more commonly used nonhierarchical methods in separating known actives and inactives [41]. 

For the second task, it is desirable to apply and test, in addition to grid based methods. Smooth Particle and Dissipative Particle methods in the spirit of [37], i.e., the stress and alignment tensors should be used as local dynamic variables. The tensorial rheological model used in [38] to... 

Compared with the grid-based methods, the SPH method handles the fountain flow more easily and naturally, without mass loss. However, for injection molding simulations, the SPH method is more time consuming, and it is difficult to enforce essential boundary conditions. 

Numerical discretization methods pose an interesting consequence for fully numerical Dirac-Hartree-Fock calculations. These grid-based methods are designed to directly calculate only those radial functions on a given set of mesh points that occupy the Slater determinant. It is, however, not possible to directly obtain any excess radial functions that are needed to generate new CSFs as excitations from the Dirac-Hartree-Fock Slater determinant. Hence, one cannot directly start to improve the Dirac-Hartree-Fock results by methods which capture electron correlation effects based on excitations that start from a single Slater determinant as reference function. This is very different from basis-set expansion techniques to be discussed for molecules in the next chapter. The introduction of a one-particle basis set provides so-called virtual spinors automatically in a Dirac-Hartree-Fock-Roothaan calculation, which are not produced by the direct and fully numerical grid-based approaches. 

The multi-configuration time-dependent Hartree (MCTDH) method [48-50] is a grid-based method that is able to treat larger systems. It uses a similar wavefunction ansatz... 

In contrast with grid-based methods, there is no need to first build analytical models for the potential energy surface, which makes Gaussian-based methods easier to use. In addition, an approximate description based on a limited number of quantum trajectories can provide a more intuitive interpretation of the reaction mechanism than the evolution of a wavepacket on a grid. However, increasing the size and complexity of the system may require an even more approximate description such as semi-classical trajectories presented below. 

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