The MO-VB Wavefunction

For each pair of active occupied SCF-MI orbitals located on the fragments, a pair of virtual orbitals and Of. is determined by minimising the variational energy corresponding to the following two-configuration wavefunction  

According to the SCF-MI strategy, the virtual Oa - Of orbitals are expanded in the partitioned basis set located on their own fragment  

It follows that, if there are na and nb SCF-MI active orbitals on fragments A and B, we obtain a total of na nb optimised virtual orbitals. The spin space is described by the spin wavefunction 

During the optimisation, the virtual orbitals Oa and Ob , see Eq. (13), are left non-orthogonal. 

To determine the virtual orbitals which minimise the variational energy (15), the derivatives with respect to the basis set expansion, see Eq. (13), and the configuration coefficients, Eq. (11) are computed. Analytic gradients and second derivatives, including mixed terms, are computed and inserted into the Newton-Raphson stabilised algorithm [19]. The detailed expression of the derivatives 

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The accuracy of the MO-VB wavefunction is expected to be close to that of a full SD-CI wavefunction involving excitations to the full virtual spaces of each monomer (vertical excitations). Very recently, a new version of the MO-VB optimization scheme has been developed that is apt to guarantee that the wavefunction approaches as close as possible the full SD-CI limit, via saturation of the optimal virtual space. Explorative calculations on the very challenging helium dimer system are encouraging. 

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