Nondynamic correlated methods

Formulas 21.1 through 21.3 are designed for Hartree-Fock wave functions. There are some attempts to define similar indices using wave functions obtained via methods including electron correlation [19]. Similarly, to the situation with respect to basis set improvement, the results based on correlated wave functions do not necessarily make the qualitative picture of bonding easier to understand. An exception is when there is a significant nondynamical correlation in the system,... 

While CCSD and especially CCSD(T) are known [36] to be less sensitive to nondynamical correlation effects than low-order perturbation theoretical methods, some sensitivity remains, and deterioration of W1 and W2 results is to be expected for systems that exhibit severe nondynamical correlation character. A number of indicators exist for this, such as the T diagnostic of Lee and Taylor [64], the size of the largest amplitudes in the converged CCSD wavefunction, and natural orbital occupations of the frontier orbitals. 

On the other hand, the PAs at the para and ortho positions are manifestly more sensitive to the electron correlation treatment than the PA at the nitrogen. From the point of view of the % diagnostic (a measure for the importance of nondynamical correlation), aniline (7 =0.0113) and N-anilinium (7 =0.0096) are very similar (and basically purely single-reference), while the p- and o-protonated species exhibit very mild multireference character (0.0149 and 0.0157, respectively). Since protonation at these sites thus involves a noticeable change in 7i, the PA is expected to be more sensitive to the correlation method. 

The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a D-based theory only needs to incorporate electron correlation. It does not rely on the concept of a fictitious noninteracting system. Consequently, the scheme is not expected to suffer from the above mentioned limitations of KS methods. In fact, the correlation energy in 1-RDM theory scales homogeneously in contrast to the scaling properties of the correlation term in DPT [14]. Moreover, the 1-RDM completely determines the natural orbitals (NOs) and their occupation numbers (ONs). Accordingly, the functional incorporates fractional ONs in a natural way, which should provide a correct description of both dynamical and nondynamical correlation. 

Keywords strongly correlated electrons nondynamic correlation density matrix renormalization group post Hartree-Fock methods many-body basis matrix product states complete active space self-consistent field electron correlation... 

Let us recall, finally, that ec CCSD approaches exploit the complementarity of the Cl and CC methods in their handling of the dynamic and nondynamic correlations. While we use the Cl as an external source of higher than pair clusters, Meissner et al. [10,72-74] exploit the CC method to correct the Cl results (thus designing the CC-based Davidson-type corrections). This aspect will also be addressed below. 

Another correlation effect that is sometimes noted arises from the fact that the SCF MOs are optimized in an uncorrelated calculation and might therefore relax if correlation was included. For our purposes this effect of correlation on orbitals can be regarded (and treated) as a part of nondynamical correlation. We will now enumerate methods for treating both nondynamical and dynamical correlation they will be surveyed only briefly as all are treated in great detail elsewhere. It should be understood that there is no sharp dividing line between nondynamical and dynamical correlation, and methods for treating one will undoubtedly account in some part for the other. It is usually most efficient to use different techniques for each, however. 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

One area in which Cl methods appear to be thoroughly superior to perturbation theory is in the treatment of multireference problems problems with substantial nondynamical correlation effects. Even UHF-based single reference perturbation theory methods may not cope with some such situations, and multireference perturbation theory, despite many efforts over the years, still appears to be far from developing a general flexible approach that is competitive with MRCI. Transition-metal chemistry, in particular, is a graveyard for UHF-based MP methods. 

We may note a number of features of these results. First, nondynamical correlation is known to be much more important in the singlet state than in the triplet. Hence it is not surprising that the multireference methods perform better here than the single reference methods. Nevertheless, the errors in the single reference meth-... 

Second, where nondynamical correlation is unimportant (that is, mainly closed-shell molecules near their equilibrium geometries), size-extensive (or nearly size-extensive) treatments like CPF, CCSD, and CISD+Q perform well. However, as bonds are stretched, or in species where nondynamical correlation is important because of near-degeneracies or open-shell effects, the performance of these methods deteriorates fairly rapidly. As might be expected, the CCSD method generally deteriorates slowest. In general, low-order perturbation theory methods do not agree well... 

The GVB and SC methods provide wave functions that are, of course, much more compact than the corresponding valence—CASSCF one (e.g., only 14 spin-coupling modes for methane with the SC method, and a single one with the GVB method). Owing to this difference in size, the GVB and SC methods cannot be expected to include the totality of the nondynamical correlation, even if these two methods treat well, by definition, the left—right correlation for each bond of the molecule. Physically, this is because the various local ionic... 

The Cl methods build upon the HF wavefunction and so there is concern that limited expansions might not be able to recover enough nondynamic correlation to provide reasonable treatment of the multiconfigurational nature of the benzynes, especially for 43. Nevertheless, CISD provides reasonable results, and the CCCl results are in excellent agreement with experiment.Both 42 and 43 are predicted to be too energetic relative to 41 with CCSD, but inclusion of triples configurations using CCSD(T) lowers this separation. 

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