Economical description of electron

The situation for transition metal chemistry has been somehow different because, while the description of the electronic effects from the MM region may not be that important, the handling of covalent connections between the QM and MM regions is critical. The methods that have been most successful in handling this situation have been the closely related IMOMM [11] and ONIOM [12, 13]. These schemes provide a computationally economical and methodologically robust method of introducing the steric effects of the... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

In this section two approaches to the selection of CSF expansion spaces for MCSCF wavefunctions have been described. Most modern MCSCF methods are best suited to the a priori selection of CSFs based on orbital occupation and spin-coupling restrictions. These a priori selection approaches should be accompanied with empirical evidence (usually more extensive MCSCF or Cl calculations at selected geometries) that a balanced description of the molecular electron correlation is achieved. A reasonable approach to use for molecular systems is to begin with an RCI expansion of all the valence electrons. The less important electrons may then optionally be described with the more restrictive, and more economical, PPMC expansion. The description of the more important electrons in the RCI expansion may be generalized, if necessary, to other direct product type expansions, thereby allowing a more thorough treatment of the correlation of these electrons. 

The theme of this chapter focuses on the possibility of combining in a practical and economic way essential conceptual and formal elements of the description of unstable states (also referred to here as nonstationary, as decaying, or as resonance states) in atoms and molecules, with computational methodologies that handle the many-electron problem (MEP) for a variety of electronic structures, especially of excited atoms. 

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