Brueckner correction

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

We present the three contributions to the Breit interaction, Brpa, which includes the first-order term together with higher-order RPA corrections, the residual second-order Breit correction, and the third-order Brueckner correction for 2s and 2p states of Uthium-hke ions, in Fig. 10. 

Relativistic corrections. Before the possible effects of TBF are examined, one should introduce relativistic corrections in the preceding nonrelativistic BHF predictions. This is done in the Dirac-Brueckner approach [4], where the nucleons, instead of propagating as plane waves, propagate as spinors in... 

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. ... 

R. Kobayashi, R. D. Amos, and N. C. Handy, Chem. Phys. Lett., 184, 195 (1991). The Analytic Gradient of the Perturbative Triple Excitations Correction to the Brueckner Doubles Method. 

While there are a large number of corrections from the next order, two particularly important ones are called Random Phase Approximation (RPA) and Brueckner orbital (BO) corrections, given by... 

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... 

We next replace the valence HF orbitals by chained Brueckner orbitals in an RPA calculation of the transition amplitude, accounting for a set of fourth-order corrections that correspond to core shielding of the Brueckner orbital corrections. This leads to a further modification of the amplitude... 

Intuitively designed damped gradient corrections have been also used to improve the EDA for correlation. The first attempt of this kind was made by Ma and Brueckner [121] in their paper on the exact second-order expansion of Ec[p]. where they also propose the functional... 

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 approximate Brueckner orbitals, (py + Spy diflFer from Hartree-Fock orbitals py in that the peaks in the radial wave functions are drawn in toward the nucleus by the attractive polarizability force. The attractive character of this force is illustrated in Fig. 8, where we plot the radial density of 2s valence electron in neutral lithium in the HF approximation together with the modification caused by the Brueckner-orbital correction. 

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). 

The best overlap orbitals are not the same as the natural spin orbital, but are probably very close (choosing the best unitary transformation of the best overlap orbitals). In a Cl expansion, the singly substituted determinants disappear if the best overlap orbitals of the same Cl problem are used. The best overlap orbitals are also called Brueckner orbitals (BO), after the American physicist Keith Brueckner, or are referred to as correlation corrected orbitals. 

It is believed [78] that a similar order-of limits problem exists for / , in such a way that the combination of Sham s exchange coefficient with the Ma-Brueckner [69] correlation coefficient yields the correct gradient expansion of in the slowly-varying high-density limit. 

Numerical tests of these gradient expansions for atoms show that the second-order gradient term provides a useful correction to the Thomas-Fermi or local density approximation for Tg, and a modestly useful correction to the local density approximation for E, but seriously worsens the local spin density results for E and E xc. In fact, the GEA correlation energies are positive The latter fact was pointed out in the original work of Ma and Brueckner [69], who suggested the first generalized gradient approximation as a remedy. 

BD(T) = Brueckner doubles CCSD(T) = coupled cluster HLC = higher-level correction QCISD(T) = quadratic configuration interaction. 

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