Basis set extension

Traditionally, the G3 energy is written in terms of corrections (basis set extensions and correlation energy contributions) to the MP4/d energy. Alternatively, the G3 energy can be specified in terms of HF and perturbation energy components. Denoting the second-, third-, and fourth-order contributions from perturbation theory by E2, E3, and E4, respectively, and the contributions beyond fourth order in a QCISD(T) calculation by EAqci, the G3 energy can be expressed as... 

Modern many-body methods have become sufficiently refined that the major source of error in most ab initio calculations of molecular properties is today associated with truncation of one-particle basis sets e.g. [1]- [4]) that is, with the accuracy with which the algebraic approximation is implemented. The importance of generating systematic sequences of basis sets capable of controlling basis set truncation error has been emphasized repeatedly in the literature (see [4] and references therein). The study of the convergence of atomic and molecular structure calculations with respect to basis set extension is highly desirable. It allows examination of the convergence of calculations with respect to basis set size and the estimation of the results that would be obtained from complete basis set calculations. 

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. 

VB model, though successful for the interactions between monovalent atoms, breaks down when 71 bonds are considered. The aim of this chapter is to bring a quantitative answer to a question which can be so summarized What is the nature of the driving force which makes benzene more stable in a D6h geometry than in an alternated Dih geometry of Kekule type Exactly the same type of question applies to the allyl radical which will also be investigated and will allow the study of the effects of configuration interaction (Cl) and basis set extension. 

Let us first use Eq. (1) to estimate the resistance to distortion arising from n interactions in the high spin states. The results are displayed in Table 2, and show a remarkable constancy at the three levels of computations used in the case of allyl. Note that when Cl is performed in the n space, the n component A of the distortion energy is not any more given by the three first terms in Eq. (1), but one can still compute it as the difference between the total distortion energy and its a component, calculated from the last three terms of Eq. (1). It turns out that A in the quadruplet allyl is effectively smaller than our higher limit of 0.5 kcal/mol, and does not appear to be sensitive to Cl or basis-set extension. Similar results are also observed in benzene and cyclobutadiene, in which the n systems of the high spin states also prove to be rather insensitive to the distortion. 

G2(MP2) theory is a variation of G2 theory that uses reduced orders of Moller-Plesset perturbation theory.76 In this theory the basis set extension corrections of G2 theory in steps 4a, 4b, and 4d are replaced by a single correction obtained at the MP2 level with the 6-311+G(3df,2p) basis set, A (+3df,2p), as given by step 4 (d) in Table 4. The total G2(MP2) energy is thus given by... 

The splitting factor of the d-polarization functions for the 3df basis set extension is 3 rather than the factor of 4 used for first- and second-row atoms. The 3d core orbitals and Is virtual orbitals are frozen in the single-point correlation calculations. 

Variations of G3 Theory At least two variations of G3 theory have been proposed. The first does the basis set extensions at the second-order Mailer-Plesset level. This method, referred to as G3(MP2) theory,97 has an average absolute deviation from experiment of 1.30 kcal/mol for the G2/97 test set and 1.18 kcal/mol for the subset of 148 neutral enthalpies (see Table 5). This is a significant improvement over the related G2(MP2) theory. The new method provides significant savings in computational time compared to G3 theory (see Figure 2). The modification to step 4 in G3 theory is shown in Table 6, along with the new higher level correction parameters. 

As shown in ref. [22], the effect of basis set extension is clearly visible the height of the peaks increased by 0.18 e A-3 for the C3=C4 double bond (Figure 6d), by 0.25 e A-3 for the C-N bonds, and by 0.24 e A-3 for C=0 bonds. For the C=0 region, extended basis sets reduce the depopulation region close to the oxygen atom, increase the peak height, displace it by 0.16 A towards the oxygen atom, and reduce the lone pairs accumulation by 0.25 e A-3. 

The LCAO approximation (2.1) may be taken as the expansion of a function as a series. As for any method which contains such an expansion, it may be assumed that upon augmenting the expansion by some other functions one arrives at a superior description of atomic (or molecular) orbitals because the number of variational parameters is increased. Examine now a basis with a doubled number of functions, i.e. a basis set in which each atomic orbital is represented by two functions. The result of this basis set extension is presented in Table 2,1 for the atoms from the first to fourth rows of the periodic system. The results for the fifth-row atoms are presented in Table... 

Increasing the basis set to 6-311G(2d,2p) or TZ(2d,2p) gives changes of up to 5 kcal/ mol, while addition of a set of f- and d-functions to form the 6-311G(2df,2pd) causes changes of 3 kcal/mol. It is clear that further basis set extension may cause changes of a few kcal/mol. 

The term efSE represents a basis set extension (BSE) effect calculated at the MP2-level. As the dimension of the truncated orbital space is increased, the magnitude of the term e SE is reduced. 

As in the two-electron case, the superscript denotes a quantity calculated in terms of the truncated set of NOs. The basis set extension effect (BSE) is calculated at the MP2-level. 

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