Closed shell bonding

The inequality X,il < 1 holds for all closed shell bonding interactions, and Xi /X3 is related to the rigidity of the bond path. This can be seen from the fact that these interactions (all the bonding interactions in this paper are closed shell) are dominated by the contraction of charge away from the inter-atomic surface towards each of the respective atomic basins. 

We have seen that the closed-shell bonding and antibonding configurations each provide an uncorrelated description of the electronic system but that a superposition of these configurations introduces correlation. In the ground state, the electrons tend to be located around opposite nuclei, whereas, in the excited state, there is a tendency for the electrons to be located around the same nucleus. Further insight into the correlation problem may be obtained by isolating the pure covalent and ionic states, where the electrons are either always located around opposite nuclei or always located around the same nucleus. 

In general, a truly uncorrelated many-particle state is always represented by a product of one-particle functions. Conversely, any superposition of such products represents a stale where the motion of the particles is correlated. Nevertheless, a Slater determinant - which leiniesents an antisymmetric superposition of spin-orbital products - may in some cases represent a tmly uncorrelated electronic state. Thus, in the closed-shell bonding and antibonding configurations of the hydrogen molecule, the Pauli principle is satisfied by an antisymmetrization of the spin part of the wave function... 

One can note some interesting features from these trajectories. For example, the Mulliken population on the participating atoms in Figure 1 show that the departing deuterium canies a full electron. Also, the deuterium transferred to the NHj undergoes an initial substantial bond stretch with the up spin and down spin populations separating so that the system temporarily looks like a biradical before it settles into a normal closed-shell behavior. 

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

I he results of their calculations were summarised in two rules. The first rule states that at least one isomer C with a properly closed p shell (i.e. bonding HOMO, antibonding I. U.MO) exists for all n = 60 - - 6k (k = 0,2,3,..., but not 1). Thus Qg, C72, Cyg, etc., are in lhi-< group. The second rule is for carbon cylinders and states that a closed-shell structure is lound for n = 2p(7 - - 3fc) (for all k). C70 is the parent of this family. The calculations Were extended to cover different types of structure and fullerenes doped with metals. 

The covalent, or shared electron pair, model of chemical bonding was first suggested by G N Lewis of the University of California m 1916 Lewis proposed that a sharing of two electrons by two hydrogen atoms permits each one to have a stable closed shell electron configuration analogous to helium... 

Irons of benzene are distributed in pairs among its three bonding tt MOs giving a closed shell electron configuration All the bonding orbitals are filled and all the electron spins are paired... 

In addition to diamond and amorphous films, nanostructural forms of carbon may also be formed from the vapour phase. Here, stabilisation is achieved by the formation of closed shell structures that obviate the need for surface heteroatoms to stabilise danghng bonds, as is the case for bulk crystals of diamond and graphite. The now-classical example of closed-shell stabilisation of carbon nanostructures is the formation of C o molecules and other Fullerenes by electric arc evaporation of graphite [38] (Section 2.4). 

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. 

At the equilibrium inter-atomic distance R, two paired electrons of occupy the bonding orbital with a closed-shell low-spin singlet (S = 0). When the bond length is further increased, the chemical bond becomes weaker. The dissociation limit of corresponds to a diradical with two unpaired electrons localized at each atom (Fig. 1). In this case, the singlet (S spin-antiparaUel) and triplet (T spin-parallel) states are nearly degenerate. Different from such a pure diradical with... 

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