Reduced basis

Various techniques exist that make possible a normal mode analysis of all but the largest molecules. These techniques include methods that are based on perturbation methods, reduced basis representations, and the application of group theory for symmetrical oligomeric molecular assemblies. Approximate methods that can reduce the computational load by an order of magnitude also hold the promise of producing reasonable approximations to the methods using conventional force fields. 

In this section we consider how Newton-Raphson iteration can be applied to solve the governing equations listed in Section 4.1. There are three steps to setting up the iteration (1) reducing the complexity of the problem by reserving the equations that can be solved linearly, (2) computing the residuals, and (3) calculating the Jacobian matrix. Because reserving the equations with linear solutions reduces the number of basis entries carried in the iteration, the solution technique described here is known as the reduced basis method. ... 

The basis entries corresponding to these two cases are given by the reduced basis, ... 

Fig. 4.4. Comparison of the computing effort, expressed in thousands of floating point operations (Aflop), required to factor the Jacobian matrix for a 20-component system (Nc = 20) during a Newton-Raphson iteration. For a technique that carries a nonlinear variable for each chemical component and each mineral in the system (top line), the computing effort increases as the number of minerals increases. For the reduced basis method (bottom line), however, less computing effort is required as the number of minerals increases.

Fig. 4.5. Convergence of the iteration to very small residuals in the reduced basis method, for systems contrived to have large positive and negative residuals at the start of the iteration. The tests assume Debye-Huckel activity coefficients ( ) and an ideal solution in which the activity coefficients are unity (o).

As we have noted, the mole numbers of minerals in the system appear as linear terms in Equation 4.5. For this reason, these equations are omitted from the reduced basis. After the iteration is complete, the values of, when unknown, are calculated according to,... 

A speciation calculation including one of the sorption models described above, or a combination of two or more sorption models, can be evaluated numerically following a procedure that parallels the technique described in Chapter 4. We begin as before by identifying the nonlinear portion of the problem to form the reduced basis,... 

To calculate a fixed activity path, the model maintains within the basis each species At whose activity at is to be held constant. For each such species, the corresponding mass balance equation (Eqn. 4.4) is reserved from the reduced basis, as described in Chapter 4, and the known value of a, is used in evaluating the mass action equation (Eqn. 4.7). Similarly, the model retains within the basis each gas Am whose fugacity is to be fixed. We reserve the corresponding mass balance equation (Eqn. 4.6) from the reduced basis and use the corresponding fugacity fm in evaluating the mass action equation. 

Data for pure hydrocarbon gases such as those presented in Figures 3-2, 3-3, and 3-4 have been put on a reduced basis and are given as Figure 3-6.4... 

W.A. Sokalski et al., An efficient procedure for decomposition of the SCF interaction energy into components with reduced basis set dependence. Chem. Phys. Lett. 153, 153-159 (1988)... 

The model proposed by Stasyuk et al. [135] describes the chain of hydrogen bonds connecting by ionic groups. The model is a development of their previous pseudo-spins reduced basis model [138], which took into account only the motion of a proton along the hydrogen bond. Reference 135 discusses the orientational degrees of freedom, which make it possible to include rotations... 

In this section, we go beyond the perturbative treatment and calculate the eigenstates of large excitonic systems by diagonalizing the Hamiltonian operator on an efficient reduced basis set [164]. 

Enhancement of two-photon cross-sections by two-dimensional and three-dimensional arrangements of monomers has been demonstrated with fluorene V-shapes and dendrimeric structures. Such multidimensional structures lead to lower two-photon absorptions than Hnear oUgomers, but they have better one-photon transparencies. An accurate calculation of large exitonic systems is obtained by diagonalizing the Hamiltonian operator on a reduced basis set. 

Most SEMO methods are based on molecular theory and occupy a reduced basis set for only valence electrons. Electron correlation is treated openly only if this is required for a suitable zero-order sketch. 

The alternative strategy is to start by inspecting the MOs obtained from a ground-state self-consistent field (SCF) computation. The major difficulty here is that the virtual orbitals of an extended basis SCF computation are completely unsuitable for the representation of an excited state. Thus, one should start from an SCF computation with a one-electron basis that does not have diffuse components (STO-3G or 3-21G), from which the orbitals can be visually inspected and an active space chosen. When the active space is correct, the reduced basis-set computation may be used as a starting point for an extended basis-set computation. 

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