Three-center bonding analyzed from correlated and uncorrelated third-order reduced density matrices

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Several approximations to the third-order reduced density matrices (3-RDM) are applied to compute approximate three-center electron sharing indices (3c-ESI) and average number of three particles (3-AN) for correlated and uncorrelated wave functions with different atomic partition schemes. Exact and approximated 3c-ESI are calculated for a set of molecules with three-center two-electron (3c-2e) and three-center four-electron (3c-4e) bonding. Results show that 3c-2e bonding is associated with positive values of 3c-ESI irrespective of the method of calculation, atomic partition employed, and approximation to the third-order density matrix used. Single-determinant calculations yield negative 3c-ESI values for 3c-4e bonds, whereas the exact CASSCF 3c-ESIs are positive or close to zero. Some approximations to the 3-RDM preserve the negative sign of the 3-ESI for 3c-4e bonds, however they perform poorly on the calculation of 3-AN. The adequacy of the 3-RDM approximation to calculate 3-AN is also analyzed, revealing that Valdemoro's approximation to the 3-RDM is the best approximation while Mazziotti's and natural-orbital based approximations yield the lowest maximum errors ​
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