Choice of basis

The first decision to be made in constructing a geochemical model is how to choose the basis, the set of thermodynamic components used to describe composition. Thermodynamics provides little guidance in our choice. Given this freedom, we choose a basis for convenience, subject to three rules  

The third rule is, in fact, a logical consequence of the first and second, but we write it out separately because it provides a useful test of a basis choice. 

The way we select components to make up the basis is similar to the way a restaurant chef might decide what foodstuffs to buy. The chef needs to be able to prepare each item on the menu from a pantry of ingredients. For various reasons (to simplify ordering, account for limited storage, minimize costs, allow the menu to be changed from day to day, and keep the ingredients fresh), the chef keeps only the minimum number of ingredients on hand. Therefore, the pantry contains 

A straightforward way to choose a basis is to select elements as components. Accounting for redox reactions, the basis also includes the electron or some measure of oxidation state. Clearly, this choice satisfies the three rules mentioned, since any species or phase is composed of elements, and reactions converting one element to another is the stuff of alchemy or nuclear physics, both of which are beyond the scope of this book. 

Throughout this book, we will choose the following species and phases as components  

The aqueous species included in the basis are known as basis species, while the remaining species in solution comprise the set of secondary species. 

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The reconstruction algorithm proposed in this work is based on a special choice of basis flinctions to expand the unknown refractive index profile. The following set of functions is used here ... 

We now show what happens if we set up tire Hamiltonian matrix using basis functions i ), tiiat are eigenfiinctions of Fand with eigenvalues given by ( equation A1.4.5) and (equation Al.4.6). We denote this particular choice of basis fiinctions as ij/" y. From (equation Al.4.3). (equation A1.4.5) and the fact that F is a Hemiitian operator, we derive... 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

In any case, the argument ignores the fact that molecular HF calculations are invariably done at the HF-LCAO level and the choice of basis set often turns out to be the dominant source of error. 

The principal distinction between various computational techniques reduces eventually to the choice of basis functions [Pg.140]

The choice of basis set in ab initio calculations has been the subject of numerous theoretical studies. Early SCF calculations utilized mainly spht-va-lence basis sets such as 3-21G and 4-31G. The importance of inclusion of d polarization functions on sulfur atoms has been stressed by several authors. For instance, Suleimenov and Ha found that the omission of d polarization functions leads to a substantially lower barrier for the internal rotation ( 16 kj mol for the central bond of H2S4) and produces an unreahstically large S-S bond length for the most stable rotamer [4]. In general, the use of... 

For practical implementations it is necessary to represent the molecular electronic wave functions as a linear combination of some convenient set of basis functions. In principle any choice of basis set is permissible although the basis set must span any electronic configuration of the molecule. This implies that the basis must form a complete set. 

Linear PCR can be modified for nonlinear modeling by using nonlinear basis functions 0m that can be polynomials or the supersmoother (Frank, 1990). The projection directions for both linear and nonlinear PCR are identical, since the choice of basis functions does not affect the projection directions indicated by the bracketed term in Eq. (22). Consequently, the nonlinear PCR algorithm is identical to that for the linear PCR algorithm, except for an additional step used to compute the nonlinear basis functions. Using adaptive-shape basis functions provides the flexibility to find the smoothed function that best captures the structure of the unknown function being approximated. 

As indicated above there may be many equivalent matrix representations for a given operation in a point group. Although the form depends on the choice of basis coordinates, the character is Independent of such a choice. However, for each application there exists a particular set of basis coordinates in terms of which the representation matrix is reduced to block-diagonal form. This result is shown symbolically in Fig. 4. ft can be expressed mathematically by the relation... 

Here, I as given by the direct sum, is a (reducible) representation of a given operation, R, Its trace is the character, a quantity that is independent of tire choice of basis coordinates. As xr is merely the sum of the diagonal elements of T, it is also equal to the sum of the traces of the individual submatrices... 

However, the values in Tables 26-30 are directly calculated BDEs, according to equation 1. Taking only those molecules for which there are reliable experimental information (Table 26), and excluding the problematic (S 2) cases (CCH-, COOH-, NCO-, CN-, NC-, CHCH2- and N3 ) and the decomposed radical (CH3CO2 ), the average MP2 error for CH3—Y is 2.0 kcalmol-1 (21 cases) and for MP4 is 3.8 kcal mol-1 (19 cases). It thus seems that the particular choice of basis set and level of calculation used here gives the best cancellation of errors for the MP2 method. 

The set of components used in a geochemical model is the calculation s basis. The basis is the coordinate system chosen to describe composition of the overall system of interest, as well as the individual species and phases that make up the system (e.g., Greenwood, 1975). There is no single basis that describes a given system. Rather, the basis is chosen for convenience from among an infinite number of possibilities (e.g., Morel, 1983). Any useful basis can be selected, and the basis may be changed at any point in a calculation to a more convenient one. We discuss the choice of basis species in the next section. 

This choice of basis follows naturally from the steps normally taken to study a geochemical reaction by hand. An aqueous geochemist balances a reaction between two species or minerals in terms of water, the minerals that would be formed or consumed during the reaction, any gases such as O2 or CO2 that remain at known fugacity as the reaction proceeds, and, as necessary, the predominant aqueous species in solution. We will show later that formalizing our basis choice in this way provides for a simple mathematical description of equilibrium in multicomponent systems and yields equations that can be evaluated rapidly. 

A determined by X-ray crystallography [56], whereas the distance to the amine groups deviates less, 2.09 A in this work and 2.01 A experimentally. Previous theoretical studies by Carloni et al. [63] and Zhang et al. [64] yielded Pt-Cl distances of 2.34 and 2.37 A, respectively. The difference from the more recently obtained value is related to the choice of basis set and exchange-correlation functional. 

The fact that the GHOs, when optimised in a molecular environment, generate an atomic orbital basis which has just the symmetry of the molecule (i.e. not higher or lower symmetry) is extremely encouraging — it tends to reassure that the GHO basis does refelct the physics of the interactions adequately. Thus, within the approximation scheme of a minimal GHO basis, the symmetry dilemma is actually a choice of basis dilemma and can be satisfactorily solved by using near-optimum scale factors for the GHOs. 

At this point we should mention that we encountered instability problems in the linear response calculations for some of the MCSCF wavefunctions at intemuclear distances larger than R—S a.u. We believe those instabilities to be artifacts of the calculations because their existence or position depends on the choice of basis set, active space or number of electrons allowed in the RAS3 space. This implies that even though it might not be possible to generate... 

One choice of basis function, based on a quadrilateral patch, is illustrated in Figure 15.2c. In the figure the element in the fth row andyth column of the mesh is assumed to have a magnitude that varies within the patch the derivative properties may be important as well. The choice of fifix, y) is not arbitrary it is made to reflect certain mathematical qualities derived, perhaps, from prior knowledge of the general behavior of similar systems as well as properties that simplify the solution process to follow. One immediately practical constraint is that the fifix, y) must satisfy the boundary conditions. Another property is that the patches meet smoothly at the intersections this is usually obtained by continuity of fifix, y) to first and second order in the derivatives. It is also convenient in many applications to choose combinations of products of functions separately dependent on x and y, reminiscent of the analytic solution, Eq. (15.2). 

With the topological analysis of the total charge density, the distinction between a covalent and a closed-shell ionic interaction can be based on the value of the Laplacian and its components at the bond critical point. Such an analysis will be most conclusive when done on a series of related compounds, analyzed with identical basis sets, as the topological values of the model density from experimental data have been found to be quite dependent on the choice of basis functions. 

In this form, it becomes possible to analyze the merits of Mulliken charge distributions in comparisons with physical observables. Namely, we want to learn the true value of n and the appropriate value of A for given choices of basis sets. Three approaches were followed to this end ... 

Calculated CX stretching frequencies for these compounds (repeating the data in Appendix A7) are provided in Table 7-3 and compared to measured values. As expected, limiting (6-311+G basis set) Hartree-Fock frequencies are all larger than experimental values. In fact, with the sole exception of methyl chloride at the 3-2IG level, Hartree-Fock frequencies are always larger than experimental frequencies, irrespective of choice of basis set. 

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