Basis set extension correction

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. 

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... 

TABLE 4. Singlet-triplet splitting AEst (in kcal/mol) in alkyl-substituted car-benes. Calculations are carried out with a correlation consistent (cc-pVTZ) basis set. Extensivity corrections are applied both a posteriori (designated by the subscript ap ) and iteratively (designated by the subscript it. ). 

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... 

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. 

In this particular case, because the molecule is nonpolar, basis set extension beyond cc-pVTZ hardly affects the bond angle. Use of empirical bond length corrections results in bond lengths that lie all within about 0.001 A of each other (Table 5). 

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

Among the most widely used ab initio methods are those referred to as Gl" and 02." These methods incorporate large basis sets including d and / orbitals, called 6-311. The calculations also have extensive configuration interaction terms at the Moller-Plesset fourth order (MP4) and fiirther terms referred to as quadratic configuration interaction (QCISD). ° Finally, there are systematically applied correction terms calibrated by exact energies from small molecules. 

As an example of the interest to scrutinise the UHF solution, one may quote the Bea problem [19]. The bond is weak but it takes plaee at short interatomic distance and is definitely not the dispersion well which one might expect from two closed shell atoms (and which occurs in Mga and heavier eompounds). Quantum chemical calculations only reproduce this bond when using large basis sets and extensive Cl calculations [20]. It is amazing to notice that the UHF solution gives a qualitatively correct behaviour, and suggests a physical interpretation of this bond since in... 

Our results indicate that the basis set i cannot describe correctly MgH- and that the ma etic susceptibility of this anion is strongly depending on the inclusion of diffuse orbitals in the basis set. We notice that the basis set 11 permits to obtain reliable results, its further extension by extra diffuse functions (basis set 111) leading approximately to the same results. MgH- should be diamagnetic, and its mean susceptibility is of the order of -22. 10-6 erg.G-2.mol". ... 

These single reference-based methods are limited to cases where the reference function can be written as a single determinant. This is most often not the case and it is then necessary to use a multiconfigurational approach. Multrreference Cl can possibly be used, but this method is only approximately size extensive, which may lead to large errors unless an extended reference space is used. For example, Osanai et al. [8] obtained for the excitation energy in Mn 2.24 eV with the QCISD(T) method while SDCI with cluster corrections gave 2.64 eV. Extended basis sets were used. The experimental value is 2.15 eV. 

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