Brueckner equation

With the NN and NY potentials, the various G matrices are evaluated by solving numerically the Brueckner equation, which can be written in operato-rial form as [13, 15]... 

A slightly improved form of this equation is the renormalized Davidson correction, which is also called the Brueckner correction ... 

Abstract We discuss the high-density nuclear equation of state within the Brueckner-Hartree-Fock approach. Particular attention is paid to the effects of nucleonic three-body forces, the presence of hyperons, and the joining with an eventual quark matter phase. The resulting properties of neutron stars, in particular the mass-radius relation, are determined. It turns out that stars heavier than 1.3 solar masses contain necessarily quark matter. 

In the framework of the Brueckner theory a rigorous treatment of TBF would require the solution of the Bethe-Faddeev equation, describing the dynamics of three bodies embedded in the nuclear matter. In practice a much simpler approach is employed, namely the TBF is reduced to an effective, density dependent, two-body force by averaging over the third nucleon in the medium,... 

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. ... 

Now we turn to the evaluation of fourth- and higher-order corrections. The largest of these is the correction that arises when the approximate Brueckner orbitals obtained by solving Eq. (74) for 6(f>v are replaced by the chained Brueckner orbitals determined by solving the second-order quasiparticle equation... 

To this end, the approach taken below is to start with an exact expression for the wave function y and the energy of a many-electron system. Such an expression is, of course, equivalent to the original Schrodinger equation, but is in such a detailed form that various correlation effects, etc., are explicit in it. From this, major correlation effects are isolated, but a means of estimating everything that is left over is also given. Semi- and non-empirical theories, then, differ only in the means by which their major parts are calculated. This approach also shows the connection between different theories one looks at what portions of the exact p or E give the Brueckner method, say. 

This is similar to the closed form of the Bethe-Goldstone equation for finite nuclei. The latter is, however, for particles in the Brueckner sea" instead of the H.F. sea represented by the Vf of Eq. (103). A Bethe-Goldstone type equation in a form which would be useful for actual calculations on finite nuclei, the properties of its solutions and its variational principle are given in the Appendix of Reference 9b. 

The remaining third-order term B q is that associated with approximate Brueckner orbitals. Note that the solution to the approximate Brueckner orbital equation (153) can be written... 

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. 

In the Brueckner approach to the nuclear many-body problem, a G-matrix is calculated which is the solution of the equation... 

The 1960s saw the applications of the many-body perturbation theory developed during the 1950s by Brueckner [13], Goldstone [38] and others to the atomic structure problem by Kelly [63-72]." These applications used the numerical solutions to the Hartree-Fock equations which are available for atoms because of the special coordinate system. Kelly also reported applications to some simple hydrides in which the hydrogen atom nucleus is treated as an additional perturbation. 

CCSD is the only pure CC method that can be used in routine applications. Explicit treatment of triples (CCSDT) is usually too expensive. However, the contribution of triples can be estimated perturbatively in the CCSD(T) method. Brueckner (B) theory is a variation of CC theory which uses orbitals that make the singles contribution vanish. The accuracy and computational cost of BD is comparable to that of CCSD. Excited electronic states may be treated within the CC formalism by the equation-of-motion (EOM-CC) approach. 

B-CC = Brueckner CC CCD = coupled-cluster doubles CCSD = coupled-cluster singles and doubles CCSDT = coupled-cluster singles, doubles, and triples CCSDTQ = coupled-cluster singles, doubles, triples, and quadruples Cl = configuration interaction EOM = equation-on-motion FCI = full-configuration interaction. 

Brueckner was the first to show for the lower orders of PT that all terms having a nonlinear dependence on N mutually cancel each other. Both Eq Eq and E have only terms linear in N, which is the reason why HF and MP2 are size-extensive. This can be shown by adding stepwise electron pairs to a closed-shell system with A o electrons. For example, the zeroth-order energy increases in this case according to equation (58) ... 

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. 

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed the Unked and unlinked equations of coupled-clustCT theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter. 

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