Calculation of coulomb integrals in molecules over an spd basis of STO
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A spherical tensor expansion of 1/r12 (where r12 is the separation considered) is used to represent Coulomb integrals in a molecular context for an MNDO method over an spd basis of STO. This is rendered invariant of the space fixed axes chosen using Wigner rotation matrices which transform the integrals from the molecular frame (as distinct from Dewar )-This procedure is found to be rigorous only in the long range limit but is satisfactory at separations of the order of most single bond lengths (as Dewar ). The pole at Rab=0 is avoided by adding a constant to the separation, which takes the value that reproduces the Slater-Condon monocentric integrals there. Extension over the whole range is carried out using a unique multiplicative polynomial from the Legendre function expansion of 1/(R ab 2 +A2) for small Rab and an exponential decay in Rab is dictated by symmetry in the overlap region expression, which retains rotational invariance.
This calculation results in an easy evaluation of these functions and their first derivatives leading to a very rapid molecular geometry optimisation taking the d-orbitals into account in an MNDO hypothesis.
Key wordsMNDO spd basis Spherical tensors Bi-electronic integrals
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