Closed-shell principle

The electron-counting rules as consequences of the closed-shell principle... 

It is extremely important to keep in mind that any electron-counting rule is underlined by an orbital requirement which is called the closed-shell principle. This principle states that a stable molecule should have all its low energy MOs fully occupied and separated from the vacant high energy ones by a significant HOMO-LUMO gap (Figure 1.15). The larger this gap, the more stable the molecule. Indeed, the existence of this gap provides the molecule with Jahn-Teller stability (Jahn-Teller... 

Exercise 1.5. What is an electron-counting rule derived from application of the closed-shell principle to H or He Why don t you find this rule in textbooks ... 

Keywords Aluminum Capping principle Closed-shell principle Clusters Condensed clusters Copper Electron-counting rules Gallium Gold Jellium model Nanoclusters Palladium Polyhedral skeletal electron pair theory Silver Tensor surface harmonic... 

Part 2, Model Chemistries, begins with Chapter 5, Basis Set Effects. This chapter discusses the most important standard basis sets and presents principles for basis set selection. It also describes the distinction between open shell and closed shell calculations. 

Conventional presentaticsis of DFT start with pure states but sooner w later encounter mixed states and d sities (ensemble densities is the usual formulation in the DFT literature) as well. These arise, for example in formation or breaking of chemical bonds and in treatments of so-called static correlation (situations in which several different one-electron configurations are nearly degenerate). Much of the DFT literature treats these problems by extension and generalization from pure state, closed shell system results. A more inclusively systematic treatment is preferable. Therefore, the first task is to obtain the Time-Dependent Variational Principle (TDVP) in a form which includes mixed states. 

The chemical reactivity of radicals is governed of course by the same chemical principles as the reactivity of systems having closed-shell ground states. Both equilibrium and rate processes are important here. The paucity of quantitative data on equilibrium and rate constants of radical reactions, suitable from the viewpoint of the present state of the theory, prevents a more rapid development in the MO applications this difficulty, however, is not specific for open-shell systems. 

A detailed diseussion of these and other variants was given in (Ref.35). Attention must be called to the fact that these methods are not variational which causes the energies obtained with them to be lower than those obtained with the FCI method. The counterpart to this deffect is that excited states, open-shell systems, and radicals, can be calculated with as much ease as the ground state and closed-shell systems. Also, the size of the calculation is determined solely by the size of the Hilbert subspace chosen and does not depend in principle on the number of electrons since all happens as if only two electrons were considered. 

In this contribution our purpose is to review the principles and the results of the momentum space approach for quantum chemistry calculations of molecules and polymers. To avoid unnecessary complications, but without loss of generality, we shall consider in details the case of closed-shell systems. 

The electronic structure method used to provide the energies and gradients of the states is crucial in photochemistry and photophysics. Ab initio electronic structure methods have been used for many years. Treating closed shell systems in their ground state is a problem that, in many cases, can now be solved routinely by chemists using standardized methods and computer packages. In order to obtain quantitative results, electron correlation (also referred to as dynamical correlation) should be included in the model and there are many methods available for doing this based on either variational or perturbation principles [41],... 

Fig. 2.2. Energy levels for ISW and 3DHO potentials. Each shows major gaps corresponding to closed shells and the numbers in circles give the cumulative number of protons or neutrons allowed by the Pauli principle. In a more realistic potential, the levels for given (n, T) are intermediate between these extremes, in which the lower magic numbers 2, 8 and 20 are already apparent. Adapted from Krane(1987).

The phenomenon of electron pairing is a consequence of the Pauli exclusion principle. The physical consequences of this principle are made manifest through the spatial properties of the density of the Fermi hole. The Fermi hole has a simple physical interpretation - it provides a description of how the density of an electron of given spin, called the reference electron, is spread out from any given point, into the space of another same-spin electron, thereby excluding the presence of an identical amount of same-spin density. If the Fermi hole is maximally localized in some region of space all other same-spin electrons are excluded from this region and the electron is localized. For a closed-shell molecule the same result is obtained for electrons of p spin and the result is a localized a,p pair [46]. 

The XC energy represents the correction to the Coulomb energy for the self-energy of an electron in a many-electron system. The latter is due to both the direct self-energy of the electron as well as the redistribution of electronic density around each electron because of the Pauli exclusion principle and the Coulomb interaction. As an example, we now discuss the case of Fermi hole and the exchange energy in Hartree-Fock (HF) theory [16]. For brevity, we restrict ourselves to closed-shell cases. 

Principles and Perspectives of Fullerene Chemistry The closed shell systems 20 (N=2),... 

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