Brueckner methods

Valeev, E. F., Schaefer III, H. F., 1998, The Protonated Water Dimer Brueckner Methods Remove the Spurious C] Symmetry Minimum , J. Chem. Phys., 108, 7197. 

E. F. Valeev and H. F. Schaefer, The protonated water dimer Brueckner methods remove the spurious Cl symmetry minimum, J. Chem. Phys. 108, 7197-7201 (1998). 

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. 

There have been attempts to apply the Brueckner method to atoms and molecules but in atoms and molecules where the gij — Ijr f are well behaved and H.F. is a perfectly valid starting point, there is no need to use such a method, with aU its attendant difficulties, - just to include /. It will be shown in Section XIX that the //s in atoms and molecules are small, in fact often negligible. ... 

Thus, methods (c) and d) include many-electron correlations ( > 2) and their effect on through the generalized SCF procedure augmented even beyond the Brueckner method. In atoms and molecules, many-electron correlations are unimportant compared to pair correlations (see Section XVIII) their effect on then will be even less important. 

Comparatively little space will therefore be devoted to some rather recent approaches, such as the plasma model of Bohm and Pines, the two-body interaction method developed by Brueckner in connection with nuclear theory, Daudel s loge theory, and the method of variation of the second-order density matrix. This does not mean that these methods would be less powerful or less impor-... 

In conclusion, we observe that the elementary partitioning method described in this section is of value not only for numerical purposes and for estimating the remainder but also for studying theoretical problems connected with conventional perturbation theory and with Brueckner s approximation for treating many-particle systems. 

Table III displays VEDEs obtained with the Brueckner-reference methods discussed in Section 5.2 and augmented, correlation-consistent, triple- basis sets [41]. AEDEs include zero-point energy differences and relaxation energies pertaining to geometrical relaxation on the neutral s potential energy surface. The average absolute error with respect to experiment is 0.05 eV [26].

The Brueckner-reference method discussed in Section 5.2 and the cc-pvqz basis set without g functions were applied to the vertical ionization energies of ozone [27]. Errors in the results of Table IV lie between 0.07 and 0.17 eV pole strengths (P) displayed beside the ionization energies are approximately equal to 0.9. Examination of cluster amplitudes amd elements of U vectors for each ionization energy reveals the reasons for the success of the present calculations. The cluster operator amplitude for the double excitation to 2bj from la is approximately 0.19. For each final state, the most important operator pertains to an occupied spin-orbital in the reference determinant, but there are significant coefficients for 2h-p operators. For the A2 case, a balanced description of ground state correlation requires inclusion of a 2p-h operator as well. The 2bi orbital s creation or annihilation operator is present in each of the 2h-p and 2p-h operators listed in Table IV. Pole strengths are approximately equal to the square of the principal h operator coefiScient and contributions by other h operators are relatively small. 

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... 

In the optimized orbitals CCD (00-CCD or OD) model, the orbitals are optimized variationally to minimize the total energy of the 00-CCD wavefunction. This allows one to drop single excitations from the wavefunction. Conceptually, 00-CCD is very similar to the Brueckner CCD (B-CCD) method. Both 00-CCD and B-CCD perform similarly to CCSD in most cases. 

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. ... 

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