Overlap integral general equation

The simplest case arises when the orbitals at the two sites are identical or accidentally have the same energy. The overlap integral will not in general be equal to zero but often is very small and may be approximated as zero for mathematical convenience. The solution to the entitled case is easily found from equation (3.14), which becomes... 

Problem 12.2 Consider a linear ion. Go back to the general secular equations (12.3), neglect all overlap integrals and assume that atoms 1 and 3 are so far apart that Hi j, is equal to zero. Show that the energy of the lowest molecular orbital is -I- V2//i 2, and determine the orbital coefficients. Is this orbital energy consistent with the observation that the equilibrium structure of the molecule is triangular rather than linear ... 

While the general expression (equation 15) will allow any reduced overlap integral to be evaluated, it is nonetheless a very lengthy expression. Simplified formulae for many specific overlap integrals have been derived. For example, the Is-Is integral for two hydrogen atoms can be written as... 

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