Nondynamic electron correlation

Two different correlation effects can be distinguished. The first one, called dynamical electron correlation, comes from the fact that in the Hartree-Fock approximation the instantaneous electron repulsion is not taken into account. The nondynamical electron correlation arises when several electron configurations are nearly degenerate and are strongly mixed in the wave function. 

When the HF wave function gives a very poor description of the system, i.e. when nondynamical electron correlation is important, the multiconfigurational SCF (MCSCF) method is used. This method is based on a Cl expansion of the wave function in which both the coefficients of the Cl and those of the molecular orbitals are variationally determined. The most common approach is the Complete Active Space SCF (CASSCF) scheme, where the user selects the chemically important molecular orbitals (active space), within which a full Cl is done. 

However, in a large number of closed shell molecules, a single Slater determinant describes the ground state wave function fairly accurately. Even in such cases inclusion of excited state configuration results in substantial lowering of total electronic energy, and this is referred to as nondynamic electron correlation. 

The simplest ab initio approach which can be used for the characterization of excited states is the configuration interaction with single excitations from the HF reference (CIS) [21]. The CIS method can be considered as the equivalent of the ground-state HF method for excited states. It does not account for so-called nondynamical electron correlation effects associated with the near degeneracy of electronic configurations, nor does it account for so-called dynamical electron correlation effects. The CIS method is computationally cheap and robust and can easily be applied to relatively large systems. 

Nondynamical electron correlation effects are generally important for reaction path calculations, when chemical bonds are broken and new bonds are formed. The multiconfiguration self-consistent field (MCSCF) method provides the appropriate description of these effects [25], In the last decade, the complete active space self-consistent field (CASSCF) method [26] has become the most widely employed MCSCF method. In the CASSCF method, a full configuration interaction (Cl) calculation is performed within a limited orbital space, the so-called active space. Thus all near degeneracy (nondynamical electron correlation) effects and orbital relaxation effects within the active space are treated at the variational level. A full-valence active space CASSCF calculation is expected to yield a qualitatively reliable description of excited-state PE surfaces. For larger systems, however, a full-valence active space CASSCF calculation quickly becomes intractable. 

It is known that, in the MO framework, the nondynamical electron correlation is accounted for by means of a so-called CASSCF calculation, which is nothing else than a full Cl in a given space of orbitals and electrons, in which the orbitals and the coefficients of the configurations are optimized simultaneously. If the active space includes all the valence orbitals and electrons, then the totality of the nondynamical correlation of the valence electrons is accounted for. In the VB framework, an equivalent VB calculation, defined with pure AOs or purely localized hybrid atomic orbitals (HAOs), would involve all the covalent and ionic structures that may possibly be generated for the molecule at hand. Note that the resulting covalent—ionic VB wave function would have the same dimension as the valence—CASSCF one (e.g., 1764 VB structures for methane, and 1764 MO SCF configurations in the CASSCF framework). 

Despite an effective treatment of nondynamical electron correlation, MCSCF calculations carried out in feasible active spaces7 do not produce quantitatively reliable results. The reason for this is the neglect of dynamical electron correlation involving the inactive occupied and virtual orbitals. Qualitatively,... 

For a general discussion of dynamic and nondynamic electron correlation, see, for example (a) DKW Mok, R Neumann, NC Handy. J Phys Chem 100 6225-6230, 1996. (b) MA Buijse, EJ Baerends. In ED EUis, ed. Density Functional Theory of Molecules, Clusters, and Solids. Dordrecht, The Netherlands Kluwer Academic, 1995, pp 1-46. 

Ugalde [92] have proposed a separation criterion according to which "The wavefunction that maximizes the electron-electron counterbalance density contains all the nondynamical electron correlation" (Abstract of Ref. [92]). 

According to the arguments in the text, and as I repeat below, the fundamental contribution of the 3d orbitals that is revealed in the computations of [95] are in the function space of the Fermi-sea nondynamical electron correlation and not of dynamical correlation. 

G. Fogarasi, R. Liu, P. Pulay, Effect of nondynamical electron correlation on the geometries of conjugated jr-systems, J. Phys. Chem. 97 (1993) 4036. 

E. Valderrama, J.M. Mercero, J.M. Ugalde, The separation of the dynamical and nondynamical electron correlation effects, J. Phys. B 34 (2001) 275. 

Among several types of the MCSCF method, the complete active space self-consistent field (CASSCF) method is commonly used at present. In fact, it has many attractive features (1) applicable to excited state as well as the ground state in a single framework (2) size-consistent (3) well defined on the whole potential energy surface if an appropriate active space is selected. However, CASSCF takes into account only nondynamic electron correlation and not dynamic correlation. The accuracy in the energy such as excitation energy and dissociation energy does not reach the chemical accuracy, that is, within several kcal/mol. A method is necessary which takes into account both the non-dynamic and dynamic correlations for quantitative description. 

The most famous MCSCF method is the complete active-space (CAS) SCF method (Roos et al. 1980), which incorporates all possible excited CSFs in the set of specific valence orbitals. Since the CASSCF method may be the simplest way to take into account the nondynamical electron correlation, this method has been applied to a wide variety of systems from small molecules to biomolecules. However, this method still has various problems e.g., the number of excited CSFs exponentially increases as the size of active space increases, the SCF process is usually converged poorly in comparison with that of the Hartree-Fock calculation, and the electron correlation is ill-balanced due to the insufficient dynamical correlation. 

Chapter 3 reviews electron correlation, to which the highest importance has been attached in quantum chemistry, for the meaning and previous approaches to incorporate it. After describing the main cause for electron correlation, dynamical and nondynamical electron correlations are introduced to clarify the details of electron correlation. As the calculation methods for these electron correlations, this chapter briefly reviews the configuration interaction and perturbation methods for dynamical correlations and the multiconflgurational self-consistent field (SCF) method for nondynamical correlations. This chapter also mentions advanced electron correlation calculation methods to incorporate high-level electron correlations. 

The GVB method takes care of all the left-right correlation in molecules, but does not include the totality of the nondynamical electron correlation since the various local ionic situations are constrained to be equal with this method (e.g., two adjacent local ionic forms H—/—h and H—/-I— will be found to possess the same weight). Accounting for the full nondynamical correlation, requires a Complete Active Space MCSCF calculation (CASSCF,which involves all possible configurations that can be constructed within the space of valence orbitals). ITaving said that, we nevertheless emphasize that as a rule, the GVB method provides results that are much closer to CASSCF quality than to Hartree-Fock. 

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