Dynamic and nondynamic correlations

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

The generic chemical problem involving both dynamic and nondynamic correlation is illustrated in Fig. 1. The orbitals are divided into two sets the active orbitals, usually the valence orbitals, which display partial occupancies (assuming spin orbitals) very different from 0 or 1 for the state of interest, and the external orbitals, which are divided into the core (largely occupied in the target state) or virtual (largely unoccupied in the target state) orbitals. The asymmetry between... 

Figure 1. Multireference problems involve both dynamical and nondynamical correlation. The nondynamical correlation is accounted for by the CASCI/CASSCF/DMRG wavefunction, which is made of multiple configurations generated in the active space with a fixed number of active electrons. The dynamical correlation is recovered on top of the multiconfigurational reference by correlating the active orbitals with orbitals in the external space (i.e., core and virtual orbitals.)...

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. 

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. 

McGuire, M. J. Piecuch, P. Balancing dynamic and nondynamic correlation for diradical and aromatic transition states a renormalized coupled-cluster study of the Cope rearrangement of 1,5-hexadiene, J. Am. Chem. Soc. 2005,127, 2608-2614. 

B. Datta, D. Mukherjee, Treatment of quasi-degeneracy in single-reference coupled-cluster theory. Separation of dynamical and nondynamical correlation effects, Chem. Phys. Lett. 235 (1995) 31. 

X. Li and J. Paldus, Simultaneous account of dynamic and nondynamic correlations based on complementarity of Cl and CC approaches, in M.R. Hoffmann, K.G. Dyall (Eds.), Low-lying potential-energy surfaces, ACS symposium series no. 828, ACS Books, Washington, 2002, pp. 10-30. 

In addition to these technical differences, however, there are differences of content between relativistic and nonrelativistic methods. The division between dynamical and nondynamical correlation is complicated by the presence of the spin-orbit interaction, which creates near-degeneracies that are not present in the nonrelativistic theory. The existence of the negative-energy states of relativistic theory raise the question of whether they should be included in the correlation treatment. The first two sections of this chapter are devoted to a discussion of these issues. 

The energies of chemical species are compared that differ in the number of paired electrons. This happens in dissociation reactions, ionization or electron attachment processes, excitation processes, etc. In general, when radicals or biradicals are compared with closed-shell systems correlation effects are important. The larger the difference in the number of paired electrons the larger the correlation effect will be (e.g., N s N - N( S) -I- N( S), three electron pairs are uncoupled). Of course, in all these cases, both dynamic and nondynamic correlation effects have to be considered while low-order MP methods can only cover dynamic correlation effects. 

The distinction between dynamical and nondynamical correlation is not sharp. However, as a working definition, nondynamical correlation is manifested as a few configurations entering the wavefunction with sizable importance. For dynamical correlation, there are many configurations with small expansion coefficients and a single dominant configuration. 

The terms dynamical and nondynamical correlation appear to have been introduced first in Sinanoglu [2]. 

This paper provides an overview of recent trends in the development of electronic structure theory for the accurate characterization of all, or large regions, of ground and excited potential energy surfaces. Topics include the treatment of dynamical and nondynamical correlation and the calculation of nonadiabatic coupling matrix elements, as arising from spin orbit coupling and from nuclear motion. 

Furthermore, one must consider where to include spin-orbit effects in relation to both dynamical and nondynamical correlation. The most widely used approach is to construct an effective Hamiltonian for spin-orbit coupling from MRCI wavefunctions that are built from a common orbital set. The small set of wavefunctions is often sufficient to describe spin-orbit coupling between the nonrelativistic surfaces, and was used to good effect by Werner in the reaction dynamics calculations presented in this symposium. Another approach is to obtain a set of natural orbitals from MRCI calculations and use these in a Cl calculation that includes spin-orbit interaction. In this way dynamical correlation, nondynamical correlation and spin-orbit interaction are treated on the same footing, albeit with some compromise on the first of these three. 

Simultaneous Account of Dynamic and Nondynamic Correlations Based on Complementarity of Cl and CC Approaches... 

In order to overcome the shortcommings of standard post-Hartree-Fock approaches in their handling of the dynamic and nondynamic correlations, we investigate the possibility of mutual enhancement between variational and perturbative approaches, as represented by various Cl and CC methods, respectively. This is achieved either via the amplitude-corrections to the one- and two-body CCSD cluster amplitudes based on some external source, in particular a modest size MR CISD wave function, in the so-called reduced multireference (RMR) CCSD method, or via the energy-corrections to the standard CCSD based on the same MR CISD wave function. The latter corrections are based on the asymmetric energy formula and may be interpreted either as the MR CISD corrections to CCSD or RMR CCSD, or as the CCSD corrections to MR CISD. This reciprocity is pointed out and a new perturbative correction within the MR CISD is also formulated. The earlier results are briefly summarized and compared with those introduced here for the first time using the exactly solvable double-zeta model of the HF and N2 molecules. 

Another aspect of ttie just mentioned complementarity is the ability with which the Cl and CC approaches account for the dynamic and nondynamic correlations. As alluded to above, the Cl approaches are very efficient in handling of the latter, already at a low-dimensional level, while the dynamic correlation requires the inclusion of a large number of highly-excited configurations, thus making unrealistic demands on the dimensionality of the Cl matrices one has to handle. For this very reason, even at the MR level, the Cl results are invariably corrected ex post by relying on various semi-empirical Davidson-type corrections. 

When lx) is an MR-CI-type wave function, the energy expression given by eq 8 has a very simple interpretation. We know that die MR-CI-type wave function can efficiently describe the nondynamic correlation while the CC Ansatz, even at the CCSD level, can very effectively account for the dynamic correlation. Thus, by combining an MR Cl and CC Ansatze, we should be able to account for both the dynamic and nondynamic correlations. The energy 5, as given by eq 8, precisely reflects this fact, with Eci accounting for the nondynamic correlation and the second term on the right hand side, which involves the CC Ansatz, for the dynamic one. 

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