Brueckner orbital

Another approach of this kind uses the approximate Brueckner orbitals from a so-called Brueckner doubles, coupled-cluster calculation [39, 40]. Methods of this kind are distinguished by their versatility and have been applied to valence ionization energies of closed-shell molecules, electron detachment energies of highly correlated anions, core ionization... 

Hartree-Fock orbitals are used. The contributions from these clusters are exactly equal to zero when Brueckner orbitals are usedso It is now also understandable that all products containing such as t t, t t k etc. will be small. Generally, the effect of linked tri-excited clusters as well as that of linked tetra-excited clusters is small. (The calculations demonstrating these effects will be discussed in Section VI.G.) It is possible to state that from three-electron clusters upwards the contribution to the correlation energy coming from linked clusters decreases rapidly. This is also understandable due to the fact that these contributions first appear in terms of higher orders in the perturbation expansion, which is demonstrated in Table 1. From the above discussion it is possible to conclude that the most important clusters are... 

The order of perturbation at which various levels of excitation first arise is illustrated in Figure 11 for three different reference functions. In Figure 11(a), the Hartree-Fock orbitals are used to form the reference function, in Figure 11(b) the bare-nucleus model is used in zero-order, while in Figure 11(c) Brueckner orbitals are used to construct the reference function. 

Figure 11 Order of perturbation at which various blocks of the configuration mixing matrix first contribute to the energy (a) for Hartree-Fock reference function (b) for ibare-nucleus> reference function (c) for reference function constructed from Brueckner orbitals... 

There are a few minor variations on the CC methods. The quadratic configuration interaction including singles and doubles (QCISD)" ° method is nearly equivalent to CCSD. Another variation on CCSD is to use the Brueckner orbitals. Brueckner orbitals are a set of MOs produced as a linear combination of the HF MOs such that all of the amplitudes of the singles configurations ( f) are zero. This method is called BD and differs from CCSD method only in fifth order." Inclusion of triples configurations in a perturbative way, BD(T), is frequently more stable (convergence of the wavefunction is often smoother) than in the CCSD(T) treatment. 

Well-known procedures for the calculation of electron correlation energy involve using virtual Hartree-Fock orbitals to construct corresponding wavefunctions, since such methods computationally have a good convergence in many-body perturbation theory (MBPT). Although we know the virtual orbitals are not optimized in the SCF procedure. Alternatively, it is possible to transform the virtual orbitals to a number of functions. There are some techniques to do such transformation to natural orbitals, Brueckner orbitals and also the Davidson method. 

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. ... 

T. D. Crawford, T. J. Lee, and H. F. Schaefer, /. Chem, Phys, 107, 9980 (1997). Spin-Restricted Brueckner Orbitals for Coupled-Cluster Wavefunctions. 

Cluster Singles, Doubles, and Triples Calculations with Hartree-Fock and Brueckner Orbital Reference Determinants A Comparative Study. 

C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, /. Chem. Phys., 109, 4171 (1998). Energies and Analytic Gradients for a Coupled-Cluster Doubles Model Using Variational Brueckner Orbitals. Application to Symmetry Breaking in O4. 

Figure 1. Electron correlation in butadiene as a function of the strength of an electric field applied along the longitudinal (x) axis relative to the correlation energy at zero field strength. The top two curves are nearly coincident, showing the similarity between CCSD and the approximate form ACCSD (see Table I). The bottom two curves are also nearly coincident. They correspond to MP2 calculations done without correlating the carbon li orbitals and with the inclusion of correlation from these core orbitals. All the other correlation treatments were done without including core correlation effects. The middle curve is the Brueckner orbital ACCD curve.

The correlation treatments were done at the levels of second order perturbation theory, coupled cluster theory with single and double substitutions (CCSD) [55 59], an approximate form of CCSD, and that form with Brueckner orbitals (BO) [102], The approximate form [103,104], using herein the designation ACCSD or ACCD, has been shown to yield potential curves, potential surface slices, and properties very close to the corresponding CC results [104 106],... 

Although we restrict outselves in this article to the Hartree-Fock spin orbitals, it should be mentioned that Paldus and Cizek have also discussed the stability of Brueckner orbitals.24... 

When one studies the effect of correlation on one-dectron properties, the question of the independent-partide scheme with respect to which correlation is defined becomes very important. For instance, the Hartree-Fock orbitals might be replaced by Brueckner orbitals, and this might produce some changes in the expectation values of some operators. 

The Brueckner orbital variant of CC should also be mentioned. CCSD puts in all single excitation effects via the wavefunction exp(Ti + T2) o- We can instead change the orbitals ip, in Oo in this wavefunction until Tj = 0. These orbitals are called Brueckner orbitals and define a single determinant reference B instead of o that has maximum overlap with the correlated wave-function. Since B-CCD " (or BD) effectively puts in Tj, it will give results similar but not identical to those from CCSD (they differ in fifth order). For BH, the corresponding B-CCD errors are 1.81, 2.88, and 5.55, compared to 1.79, 2.64 and 5.05, for CCSD as a function of R. . See also B-CCD for symmetry breaking problems. ... 

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... 

A partial alternative to CCSD that provides an analogous singles effect within a different computational strategy was to rotate the orbitals to define Brueckner orbitals that have the property that Tj = 0. This was the route of Dykstra [55]. Making T] = 0 via orbital rotation only requires four of the 45 — 9 = 36 diagrams in CCSD to actually be evaluated, making it simpler in that respect than CCSD, but it requires repeated orbital iterations and transformations until convergence is reached. 

Table 14 Structure of the Hamiltonian matrix when the single determinantal reference function is constructed from Brueckner orbitals. The order of perturbation theory in which each block contributes to the correlation energy is indicated...

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