Many-body counterpoise correction

If, instead of dimer interactions, many-body effects are calculated, the Boys-Bemardi recipe is in a straightforward way extended to read each subsystem is to be calculated in the complete basis of the supersystem , a recipe that has been applied to trimer interactions and, albeit approximately, to cluster calculations . Wells and Wilson call this the site-site counterpoise function method and formulate it nicely for two-body, three-body, etc., interactions. Computing only a counterpoise correction for pairs of monomers and assuming additivity proved to overestimate the BSSE even for small systems . 

The best estimate obtained for the correlation contribution to the interaction energy with the many-body approach, the counterpoise-corrected result with the largest basis set (150 functions), is - 0.70 kcal/mol. This value required 9 hr of Cray X-MP time to compute even so, it recovers only some 70% of the total correlation effect. Smaller sets yield only half the true value. In summary, the authors were pessimistic about the ability of computing very accurate correlation contributions, especially for systems larger than the water dimer. 

For extensive basis sets, an optimal description of the subsystems X and Y and the supersystem X... Y will be obtained. The basis set superposition error will then be very small. In recent work. Wells and Wilson did not use the function counterpoise correction in the usual fashion described above. They pointed out not only that the Boys-Bemardi procedure overcorrects for basis set superposition effects but also that it cannot be uniquely generalized for the calculation of a many-body interaction. Wells and Wilson argue that the function counterpoise correction should be used as a test for basis set superposition errors. 

Function counterpoise correction for the ground state of the neon atom calculated using diagrammatic many-body perturbation theory and employing a systematic sequence of even-tempered basis sets of Gaussian-type functions. In this table G represents a set of ghost orbitals. The NeG internuclear separation is 5.0 bohr. ... 

In this approach it is assumed that the basis set superposition error in the many-body cluster can be approximated by the sum of the Boys-Bemardi function counterpoise corrections for pairs of bodies. Hence the total interactions for an N-body cluster using the pairwise additive function counterpoise correction is given by... 

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