Brueckner condition

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... 

At convergence, the orbitals will obey the Brueckner conditions... 

This is the case if the y>i are chosen as the best overlap 5) or Brueck-ner 34,36) spin orbitals. That such a choice is always possible has been diown by Brenig and independently by Nesbet 4). Eq. (54) is usually referred to as the Brueckner condition, in contrast to the Brillouin condition (53). Note that (53) can be regarded as either a theorem, if one defines the Hartree-Fock equation in the conventional way, or as a condition from which the conventional Hartree-Fock equation can be derived. 

Note that the definitions in Eq. (55—57) depend on whether wehave chosen the Brillouin (53) or the Brueckner condition (54), so we should, when necessary, state this choice explicitly. Numerically the differences are usually very small. 

Because (4> ff S) is not in itself a variational expression, its unconstrained minimum value is not simply related to an eigenstate of the Hamiltonian Hv defined by v in Eq.(3), whereas Eq.(2) defines F[p only for such eigenstates. Any arbitrary trial function J —> can be expressed in the form + Aca with ca = 1. If the minimizing trial function in Eq.(3) were not an eigenfunction of Hv, then for some subset of trial functions, using the Brueckner-Brenig condition,... 

Thus the anomalous term exactly compensates the correction introduced in going from e% to ep. This fact was first recognized byKohnand Luttinger. Later on, Luttinger and Ward showed that this compensation takes place at all orders, if the unperturbed Fermi sea has spherical symmetry and if the interactions are isotropic. They established in this way the correctness of the Brueckner-Goldstone perturbation formula under these special conditions. Here, however, as the electron-lattice interaction v r) is not isotropic, it is clear that such a compensation will not occur at all orders iri fact it happens accidentally for second-order terms. We thus have ... 

The single determinantal function = 4>g, which fulfills the above condition, is called a Bruckner function (O. Sinanoglu and K.A. Brueckner Three Approaches to Electron Correlation in Atoms Yale Univ. Press, New Haven and London, 1970). 

In this context, it is pertinent to recall that in many cases one can obtain the so-called best overlap orbitals [64] of DODS type which are produced by the given many-electron wave function. These orbitals were considered in [65] where they were identified with spin-polarized Brueckner orbitals. However, they exist if and only if the so-called nonsinglet Brueckner instabihty conditions are satisfied. At last, if the correct spin-projected determinant is involved in the consideration, then it is always possible to construct the best overlap orbitals of DODS type for the given exact or approximate state vector ). These orbitals were recently introduced [62] and named the spin-polarized extended Bmeckner (SPEB) orbitals. By construction, they maximize T). 

At this point, we mention that the orbital-rotation parameters may also be determined by extending the projeetion manifold to the single excitations, replacing the orbital conditions (13.8.22) by the amplitude equations (13.8.20) for the singles. This approach is called Brueckner coupled-cluster (BCC) theory [5,31,32]. In BCC theory, neither the energy nor the amplitude equations depend on the multipliers and no multipliers must be set up to obtain the BCC wave function. 

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