Three-centre integrals

Three centre integrals the two basis functions are on different nuclei and the attracting centre is different from both. This case occurs in the water molecule, for example. 

In this Section we discuss and compare the results obtained using the various theories of Section 3, and a variational transition state theory with a tunnelling correction [9]. The reaction we concentrate on is H+BrH - HBr+H. We emphasize the rate constants, but also discuss reaction cross sections. The potential energy surface we used in the H+BrH computations is of a semiempirical diatomics in molecules type, the form of which (called DIM-3C) is due to Last and Baer [ll]. The surface contains a three-centre integral term that has been parameter-ised [33] by comparing ESA-CSA calculations with an experimental [3] room temperature rate constant for the D+BrH -) DBr+H reaction. The minimum potential energy path is collinear, and there is a strong... 

In order to obtain Va (r) at point r it is sufficient to calculate the distances of the point from any of the nuclei (trivial) as well as the one-electron integrals, which appear after inserting into (14.34) p (/) =2J2i WA,i(j ) - Within the LCAO MO approximation the electron density distribution pA represents the sum of products of two atomic orbitals (in general centred at two different points). As a result the task reduces to calculating typical one-electron three-centre integrals of the nuclear attraction type (cf. Chapter 8 and Appendix P), because the third centre corresponds to the point r (Fig. 14.13). There is no computational problem with this for contemporary quantum chemistry. 

One-electron integrals involving three centres (two from the basis functions and one from the operator) are set to zero. 

The Ajg integral is just a three-centre one-electron integral, which can be evaluated analytically. The integration over coordinate 1 may then be approximated as a sum over a finite set of grid points in the physical space. 

When the number of atoms in the molecule increase above two, three- and four-centre integrals enter the expressions. Matcha and Kern115 have given expressions for these. However, there is also the problem of the number of integrals to be calculated with large molecules this will be extremely time-consuming. 

Xa F X/j) <—(Xa Jy X/j) y l(x a0 g (5.13) -leant compared to the three- and four-centre integrals, pseudospectral methods may i i nrovide energies at the same accuracy as fully analytical methods. For small to medium ... 

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