Brueckner Orbitals in Coupled Cluster Theory

In 1958, Nesbet extended Brueckner s theory for infinite nuclear mat-ter to nonuniform systems of atoms and molecules. By consideration of the CISD problem in which the electronic Hamiltonian is diagonalized within the basis of the reference and all singly and doubly excited determinants, Nesbet explained that Brueckner theory allows one to construct a set of orthonormal molecular orbitals for which the correlated wavefunction coefficients for all singly excited determinants vanish. Unfortunately, the construction of the set of orbitals that fulfill this Brueckner condition can be determined only a posteriori from the single excitation coefficients computed in a given orbital basis. As a result, the practical implementation of Brueckner-orbital-based methods has 

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma.  

The symmetric spin-orbital basis, which was also used to construct the spin-restricted (zT) correction, also provides a route to a spin-restricted open-shell B-CC theory (RB-CC). In this spin basis, the Tj amplitudes may be shown to have the symmetries 

At convergence, the orbitals will obey the Brueckner conditions 

These equations provide the basis for the RB-CC method, since they do not imply any loss of spin restriction on the molecular orbitals as the rotation is applied. Furthermore, the RB-CC method may be trivially implemented within existing ROHF-CCSD programs by a simple symmetrization of the standard (a, 3) Tj amplitudes into the new spin basis prior to the rotation. 

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