Multilevel method

In Section 6.2.6, we considered approaches to the HF limit derived under the assumption that various aspects of basis-set incompleteness (radial, angular, etc.) could be accounted for in some additive fashion (see Eq. (6.5)). In essence, multilevel methods carry tliis approach 

triple-f MP2 energies at double-1 MP2 geometries are augmented witli a correction for doubles contributions beyond second order (line 2 on die r.h.s. of Eq. (7.61)) and a correction for basis set size increase beyond triple-f (line 3 on die r.h.s. of Eq. (7.61) where the T superscript in the first basis set implies that polarization functions from cc-pVTZ were used in conjunction with valence functions from cc-pVQZ). 

A modification of G2 by Pople and co-workers was deemed sufficiently comprehensive tliat it is known simply as G3, and its steps are also outlined in Table 7.6. G3 is more accurate titan G2, witli an error for the 148-molecule heat-of-formation test set of 0.9 kcal mol . It is also more efficient, typically being about twice as fast. A particular improvement of G3 over G2 is associated with improved basis sets for tlie third-row nontransition elements (Curtiss et al. 2001). As with G2, a number of minor to major variations of G3 have been proposed to either improve its efficiency or increase its accuracy over a smaller subset of chemical space, e.g., the G3-RAD method of Henry, Sullivan, and Radom (2003) for particular application to radical thermochemistry, the G3(MP2) model of Curtiss et al. (1999), which reduces computational cost by computing basis-set-extension corrections at the MP2 level instead of the MP4 level, and the G3B3 model of Baboul et al. (1999), which employs B3LYP structures and frequencies. 

It should be noted that G2 and G3 potentially fail to be size extensive because of the correction term in step 8. If one is studying a homolytic dissociation into two components, at what point along the reaction coordinate are tlie formerly paired electrons considered to be unpaired There will be a discontinuity in tlie energy at that point. In addition, G3 theory uses a different correction for atoms than for molecules, and this too fails to be size extensive. 

A somewhat more obviously empirical variation on the multilevel approach is the multi-coefficient method of Truhlar and co-workers. Although many different variations of this approach have now been described, it is simplest to illustrate the concept for the so-called multi-coefficient G3 (MCG3) model (Fast, Sanchez, and Truhlar 1999). In essence, the model assumes a G3-like energy expression, but each term has associated with it a coefficient that is not restricted to be unity, as is the case for G3. Specifically 

A modification of G2 by Pople and co-workers was deemed sufficiently comprehensive that it is known simply as G3, and its steps are also outlined in Table 7.5. G3 is more accurate than G2, with an error for the 148-molecule heat-of-formation test set of 0.9 kcal mol-1. It is also more efficient, typically being about twice as fast. 

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Applications of the fluctuating charge model have relied on iterative methods to determine the converged charges [52, 159, 164, 196] and for very large-scale systems, multilevel methods have also been developed [197, 198], MC methods have also been used with fluctuating-charge models [116, 194],... 

In a study of 110 G1 and G2 molecules composed of G, H, O, and F, the average BAG-MP4 unsigned error in predicted heat of formation was 2.1 kcal mol (Zachariah etal. 1996). As the MP4 calculation uses a relatively modest basis set size, the BAG procedure is quite fast by comparison to some of the multilevel methods described above. On the other hand, as with any method relying on pairwise parameterization, extension to a large number of atoms requires a great deal of parameterization data, and this is a potential limitation of the BAG method when applied to systems containing atoms not already parameterized. 

Collective coordinates, 35, 98 Collision theory, 528, 542 Comparative molecular field analysis, 308-310 Complete basis set, see Multilevel methods) Compressibility, 418, 446 Condensed-phase effects, see also Solvation... 

This sort of analysis can be applied to other methods. Britz and Strutwolf [152] applied it to the BDF method using 5-point discretisation along X, and, also for 5-point approximations, Strutwolf and Britz [531] applied it to extrapolation. For a multilevel method such as BDF, the analysis results in a polynomial in , and complex roots are possible. For example, Lapidus and Pinder [350] treat the DuFort-Frankel method it results in a quadratic equation in but it is clear that is is unconditionally stable (even though we have seen that is not consistent). 

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