Non Dynamical Electron Correlation

Non dynamical electron correlation is the part of the total correlation that is taken into account in a CASSCF calculation that correlates the valence electrons in valence orbitals. Physically, the non dynamical electron correlation is a Coulomb correlation that permits the electrons to avoid one another and reduce their mutual repulsion as much as possible with respect to a given zero order electronic structure defined by the Hartree-Fock wave function. In VB terms, the non dynamical correlation ensures a correct balance between the ionic and covalent components of the wave function for a given electronic system. The dynamical correlation is just what is still missing to get the exact nonrelativistic wave function. 

The essential part of non dynamical correlation energy for polyatomic molecules is the left-right electron correlation , which is concerned with the ionic-covalent balance within a given two-electron bond. Let us therefore discuss this type of correlation. 

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In this and the following sections we will introduce the Hartree-Fock (HF) approximation and some of the fundamental concepts intimately connected with it, such as exchange, selfinteraction, dynamical and non-dynamical electron correlation. We will meet many of these terms again in our later discussions on related topics in the framework of DFT. The HF... 

Hamiltonian proposed by Muller and Plesset gives rise to a very successful and efficient method to treat electron correlation effects in systems that can be described by a single reference wave function. However, for a multireference wave function the Moller-Plesset division can no longer be made and an alternative choice of B(0> is needed. One such scheme is the Complete Active Space See-ond-Order Perturbation Theory (CASPT2) developed by Anderson and Roos [3, 4], We will briefly resume the most important definitions of the theory one is referred to the original articles for a more extensive description of the method. The reference wave function is a CASSCF wave function that accounts for the largest part of the non-dynamical electron correlation. The zeroth-order Hamiltonian is defined as follows and reduces to the Moller-Plesset operator in the limit of zero active orbitals ... 

D. Cremer, Density functional theory Coverage of dynamic and non-dynamic electron correlation effects. Mol. Phys. 99 (2001) 1899. 

DF theory has the simplicity of an independent-particle model, yet it can be applied successfully to those systems-such as transition metal complexes - where non-dynamical electron correlation is of primary importance. DF-based methods are, in general, very easy to use, no matter how sophisticated the functional employed to describe the electron correlation. Also, more sophisticated functionals do not increase the computational requirements significantly, as opposed to post Hartree-Fock ab initio calculations. The application of approximate density functional theory has been reviewed by Ziegler and others [5]. 

The electronic structure is computed using the CASSCF method. Using the standard 6-3IG basis set, we choose the 6 7t orbitals as active. The degenerate HOMO, HOMO-1 and matching degenerate LUMO, LUMO-l-1 are needed to recover the non-dynamic electron correlation. The remaining pair of benzene n orbitals contributes to dynamic correlation and has to be included for stability (because of a large dynamic correlation effect). 

Electron structure calculations often become difficult when transition metals are involved. If the system has incomplete d shells many electron configurations contribute even to the ground state leading to non-dynamical electron correlation. Wave function-based methods with multideterminant references are required for high accuracy. Density functional theory is often successful, but no current functional is reliable for transition metals compounds in general. QMC as a wave function-based method has to use multideterminant... 

In this section, we discuss in quantitative terms the fragmentation pathway of the thymine dimer radical cation [27] on the basis of calculations that take account of non-dynamical electron correlation effects. 

Further developments in F12 theory mainly address three directions (1) Combination with strategies to treat large systems with wavefunction methods, (2) balanced treatment of dynamic and non-dynamic electron correlation in multireference methods, and (3) relativistic effects. 

Through the similarity transformations and diagonalizations described, the EOM and STEOM methods incorporate both dynamic and non-dynamic electron correlation effects. The ground and excited states of a system are obtained in a uniform manner in these methods. Similarly, the singlet and triplet states are treated consistently in the DIP methods, as they are simply deletions of two electrons of appropriate spins. 

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