Gaussian product rule

Let us finally consider the Gaussian product rule for a simple product of two s functions... 

Fig. 9.2. The Gaussian product rule. The shaded areas represent the product of the two individual Gaussians. In the upper plots, the exponents are equal to 1 and the two Gaussians are centred at 5 and 1 in the lower plots, the exponents are (for the Gaussian at the origin in both plots) and 25 (for the Gaussian centred at 0 and 1).

Using the Gaussian product rule (9.2.21), this integral becomes... 

The partitioning functions have been chosen variously as Gaussians, atomic-like densities (proatoms, [30]), and inverse powers of r the latter is implemented in the DV program. One advantage of this approach, as emphasized by Becke [31], is that classical product rules can be applied to the overlapping multicenter density, to obtain integrals of high precision. 

The most frequently used basis function is the Gaussian-type function in quantum chemistry calculations. This is due to the following product rule of Gaussian-type functions (Szabo and Ostlund 1996). 

According to Baldwin s rules, the anionic 5-endo-trig cyclization is a disfavoured process, but it has been found to be facilitated by the introduction of two fluorine atoms at the terminal carbon atom. ° Extensive ab initio calculations (Gaussian 94 and Mulliken) for homoallylic alkoxides found that the highly electronegative fluorine atoms stabilized both transition state and product. 

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